reduce-historical/r38/log/trigsimp.rlg

Tue Apr 15 00:36:09 2008 run on win32
% Test file for TrigSimp package

%-------------------------TrigSimp--------------------------

trigsimp(tan(x+y), keepalltrig);


  - (tan(x) + tan(y))
----------------------
  tan(x)*tan(y) - 1

trigsimp(ws, keepalltrig, combine);


tan(x + y)

trigsimp(sin(5x-9y));


                     4       9                     4       7
 - 4096*cos(x)*sin(x) *sin(y)  + 9216*cos(x)*sin(x) *sin(y)

                     4       5                     4       3
 - 6912*cos(x)*sin(x) *sin(y)  + 1920*cos(x)*sin(x) *sin(y)

                    4                            2       9
 - 144*cos(x)*sin(x) *sin(y) + 3072*cos(x)*sin(x) *sin(y)

                     2       7                     2       5
 - 6912*cos(x)*sin(x) *sin(y)  + 5184*cos(x)*sin(x) *sin(y)

                     2       3                    2                           9
 - 1440*cos(x)*sin(x) *sin(y)  + 108*cos(x)*sin(x) *sin(y) - 256*cos(x)*sin(y)

                    7                    5                    3
 + 576*cos(x)*sin(y)  - 432*cos(x)*sin(y)  + 120*cos(x)*sin(y)

                                       5       8                     5       6
 - 9*cos(x)*sin(y) + 4096*cos(y)*sin(x) *sin(y)  - 7168*cos(y)*sin(x) *sin(y)

                     5       4                    5       2                   5
 + 3840*cos(y)*sin(x) *sin(y)  - 640*cos(y)*sin(x) *sin(y)  + 16*cos(y)*sin(x)

                     3       8                     3       6
 - 5120*cos(y)*sin(x) *sin(y)  + 8960*cos(y)*sin(x) *sin(y)

                     3       4                    3       2                   3
 - 4800*cos(y)*sin(x) *sin(y)  + 800*cos(y)*sin(x) *sin(y)  - 20*cos(y)*sin(x)

                            8                            6
 + 1280*cos(y)*sin(x)*sin(y)  - 2240*cos(y)*sin(x)*sin(y)

                            4                           2
 + 1200*cos(y)*sin(x)*sin(y)  - 200*cos(y)*sin(x)*sin(y)  + 5*cos(y)*sin(x)

trigsimp(ws, combine);


sin(5*x - 9*y)

trigsimp(cos(10x), cos);


          10              8              6             4            2
512*cos(x)   - 1280*cos(x)  + 1120*cos(x)  - 400*cos(x)  + 50*cos(x)  - 1

trigsimp(cos(10x), sin);


             10              8              6             4            2
 - 512*sin(x)   + 1280*sin(x)  - 1120*sin(x)  + 400*sin(x)  - 50*sin(x)  + 1

trigsimp((sin(x-a)+sin(x+a))/(cos(x-a)+cos(x+a)));


 sin(x)
--------
 cos(x)

trigsimp(cos(6x+4y), sin);


                        5       3                           5
256*cos(x)*cos(y)*sin(x) *sin(y)  - 128*cos(x)*cos(y)*sin(x) *sin(y)

                           3       3                           3
 - 256*cos(x)*cos(y)*sin(x) *sin(y)  + 128*cos(x)*cos(y)*sin(x) *sin(y)

                                 3
 + 48*cos(x)*cos(y)*sin(x)*sin(y)  - 24*cos(x)*cos(y)*sin(x)*sin(y)

             6       4             6       2            6             4       4
 - 256*sin(x) *sin(y)  + 256*sin(x) *sin(y)  - 32*sin(x)  + 384*sin(x) *sin(y)

             4       2            4             2       4             2       2
 - 384*sin(x) *sin(y)  + 48*sin(x)  - 144*sin(x) *sin(y)  + 144*sin(x) *sin(y)

            2           4           2
 - 18*sin(x)  + 8*sin(y)  - 8*sin(y)  + 1

trigsimp(ws, expon);


  12*i*x + 8*i*y
 e               + 1
---------------------
     6*i*x + 4*i*y
  2*e

trigsimp(ws, hyp);


                                 5          3
256*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y)

                                    5
 + 128*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y)

                                    3          3
 + 256*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y)

                                    3
 + 128*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y)

                                             3
 + 48*cosh(i*x)*cosh(i*y)*sinh(i*x)*sinh(i*y)

                                                             6          4
 + 24*cosh(i*x)*cosh(i*y)*sinh(i*x)*sinh(i*y) + 256*sinh(i*x) *sinh(i*y)

                6          2               6                4          4
 + 256*sinh(i*x) *sinh(i*y)  + 32*sinh(i*x)  + 384*sinh(i*x) *sinh(i*y)

                4          2               4                2          4
 + 384*sinh(i*x) *sinh(i*y)  + 48*sinh(i*x)  + 144*sinh(i*x) *sinh(i*y)

                2          2               2              4              2
 + 144*sinh(i*x) *sinh(i*y)  + 18*sinh(i*x)  + 8*sinh(i*y)  + 8*sinh(i*y)  + 1

trigsimp(ws, combine);


cosh(6*i*x + 4*i*y)

trigsimp(ws, trig, combine);


cos(6*x + 4*y)

trigsimp(sqrt(1-cos(2x)));


sqrt(2)*abs(sin(x))

trigsimp(sin(x)^20*cos(x)^20, sin);


      20        20            18            16             14             12
sin(x)  *(sin(x)   - 10*sin(x)   + 45*sin(x)   - 120*sin(x)   + 210*sin(x)

                       10             8             6            4            2
           - 252*sin(x)   + 210*sin(x)  - 120*sin(x)  + 45*sin(x)  - 10*sin(x)

           + 1)

trigsimp(sin(x)^20*cos(x)^20, cos);


      20        20            18            16             14             12
cos(x)  *(cos(x)   - 10*cos(x)   + 45*cos(x)   - 120*cos(x)   + 210*cos(x)

                       10             8             6            4            2
           - 252*cos(x)   + 210*cos(x)  - 120*cos(x)  + 45*cos(x)  - 10*cos(x)

           + 1)

trigsimp(sin(x)^20*cos(x)^20, compact);


      20       20
cos(x)  *sin(x)

trigsimp(sin(x)^10, combine);


  - cos(10*x) + 10*cos(8*x) - 45*cos(6*x) + 120*cos(4*x) - 210*cos(2*x) + 126
------------------------------------------------------------------------------
                                     512

trigsimp(ws, hyp);


            10
 - sinh(i*x)

trigsimp(ws, expon);


     20*i*x       18*i*x       16*i*x        14*i*x        12*i*x        10*i*x
( - e       + 10*e       - 45*e       + 120*e       - 210*e       + 252*e

         8*i*x        6*i*x       4*i*x       2*i*x             10*i*x
  - 210*e      + 120*e      - 45*e      + 10*e      - 1)/(1024*e      )

trigsimp(ws, trig);


      10
sin(x)

int(sin(x+y)*cos(x-y)*tan(x), x);


int(cos(x - y)*sin(x + y)*tan(x),x)

int(trigsimp(sin(x+y)*cos(x-y)*tan(x)), x);


       2                                                         2
 cos(x) *x - cos(x)*sin(x) - 2*cos(y)*log(cos(x))*sin(y) + sin(x) *x
---------------------------------------------------------------------
                                  2

% int(sin(x+y)*cos(x-y)/tan(x), x) hangs
int(trigsimp(sin(x+y)*cos(x-y)/tan(x)), x);


                                   x  2
(cos(x)*sin(x) - 2*cos(y)*log(tan(---)  + 1)*sin(y)
                                   2

                      x
  + 2*cos(y)*log(tan(---))*sin(y) + x)/2
                      2

trigsimp(2tan(x)*(sec(x)^2 - tan(x)^2 - 1));


0

on rationalize;


df(sqrt(1+cos(x)), x, 4);


                              4            2       2            2
(sqrt(cos(x) + 1)*( - 4*cos(x)  - 20*cos(x) *sin(x)  + 12*cos(x)

                      2                       4            2
     - 4*cos(x)*sin(x)  + 8*cos(x) - 15*sin(x)  + 16*sin(x) ))/(16

           4           3           2
   *(cos(x)  + 4*cos(x)  + 6*cos(x)  + 4*cos(x) + 1))

off rationalize;


trigsimp(ws);


 sqrt(cos(x) + 1)
------------------
        16

df(2cos((x+y)/2)*cos((x-y)/2), x);


         x - y        x + y          x + y        x - y
 - (cos(-------)*sin(-------) + cos(-------)*sin(-------))
           2            2              2            2

trigsimp(ws, combine);


 - sin(x)

df(int(1/cos(x), x), x);


          x  2
  - (tan(---)  + 1)
          2
--------------------
        x  2
   tan(---)  - 1
        2

trigsimp(ws, combine);


   1
--------
 cos(x)

trigsimp(cos(100x));


                                     100
633825300114114700748351602688*sin(x)

                                          98
 - 15845632502852867518708790067200*sin(x)

                                           96
 + 192128294097091018664344079564800*sin(x)

                                            94
 - 1505335087771022414277335056384000*sin(x)

                                            92
 + 8567473526884295537508113973248000*sin(x)

                                             90
 - 37750993877408064336851542202122240*sin(x)

                                              88
 + 134036108580690866727917044786790400*sin(x)

                                              86
 - 394078512785625681900511864396185600*sin(x)

                                              84
 + 978503372439851812055958467641344000*sin(x)

                                               82
 - 2082455895192505138478065456775168000*sin(x)

                                               80
 + 3842131126630171980492030767750184960*sin(x)

                                               78
 - 6200783636440931286187342812099379200*sin(x)

                                               76
 + 8816739233064449172547628060953804800*sin(x)

                                                74
 - 11108623702136905456127648087408640000*sin(x)

                                                72
 + 12460295938318846194767764735918080000*sin(x)

                                                70
 - 12489614281703125832873100652943769600*sin(x)

                                                68
 + 11221137831217652115471926367879168000*sin(x)

                                               66
 - 9058026923994972189597820080095232000*sin(x)

                                               64
 + 6581798018959761296303294062264320000*sin(x)

                                               62
 - 4310885252184171141438414824407040000*sin(x)

                                               60
 + 2547463753712583633893763260298035200*sin(x)

                                               58
 - 1358954443662228159129584379363328000*sin(x)

                                              56
 + 654531379770880870350000868032512000*sin(x)

                                              54
 - 284578860769948204500000377405440000*sin(x)

                                              52
 + 111631674825053695350740279623680000*sin(x)

                                             50
 - 39472960218138986676021762874933248*sin(x)

                                             48
 + 12566106098549963273941439584665600*sin(x)

                                            46
 - 3595780740528756614156758967910400*sin(x)

                                           44
 + 923024074019658505866132324352000*sin(x)

                                           42
 - 212040013118649088525828358144000*sin(x)

                                          40
 + 43468202689323063147794813419520*sin(x)

                                         38
 - 7925478751208973645460484915200*sin(x)

                                         36
 + 1280241627320751027747867852800*sin(x)

                                        34
 - 182395347175955031090266112000*sin(x)

                                       32
 + 22799418396994378886283264000*sin(x)

                                      30                                     28
 - 2485387148331694929142087680*sin(x)   + 234623135747458180159897600*sin(x)

                                    26                                   24
 - 19023497493037149742694400*sin(x)   + 1312104559685287280640000*sin(x)

                                 22                                20
 - 76111992112891822080000*sin(x)   + 3662889620432918937600*sin(x)

                               18                             16
 - 143850563845029888000*sin(x)   + 4517474603507712000*sin(x)

                            14                          12
 - 110586893598720000*sin(x)   + 2042087523840000*sin(x)

                        10                      8                    6
 - 27227833651200*sin(x)   + 246628928000*sin(x)  - 1386112000*sin(x)

                 4              2
 + 4165000*sin(x)  - 5000*sin(x)  + 1

trigsimp(ws, combine);


cos(100*x)

trigsimp(sinh(3a+4b-5c)*cosh(3a-5b-6c));


                                     5        10
16384*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                        5        8
 + 36864*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                        5        6
 + 28672*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                       5        4
 + 8960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                      5        2
 + 960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                     5
 + 16*cosh(a)*cosh(b)*cosh(c)*sinh(a)

                                        3        10
 + 16384*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                        3        8
 + 36864*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                        3        6
 + 28672*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                       3        4
 + 8960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                      3        2
 + 960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c)

                                     3
 + 16*cosh(a)*cosh(b)*cosh(c)*sinh(a)

                                               10
 + 3072*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c)

                                               8
 + 6912*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c)

                                               6
 + 5376*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c)

                                               4
 + 1680*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c)

                                              2
 + 180*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c)

                                                            5                11
 + 3*cosh(a)*cosh(b)*cosh(c)*sinh(a) + 16384*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        5                9
 + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        5                7
 + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        5                5
 + 19712*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                       5                3                      5
 + 3520*cosh(a)*sinh(a) *sinh(b)*sinh(c)  + 176*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        3                11
 + 16384*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        3                9
 + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        3                7
 + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                        3                5
 + 19712*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                       3                3                      3
 + 3520*cosh(a)*sinh(a) *sinh(b)*sinh(c)  + 176*cosh(a)*sinh(a) *sinh(b)*sinh(c)

                                       11
 + 3072*cosh(a)*sinh(a)*sinh(b)*sinh(c)

                                       9                                       7
 + 8448*cosh(a)*sinh(a)*sinh(b)*sinh(c)  + 8448*cosh(a)*sinh(a)*sinh(b)*sinh(c)

                                       5                                      3
 + 3696*cosh(a)*sinh(a)*sinh(b)*sinh(c)  + 660*cosh(a)*sinh(a)*sinh(b)*sinh(c)

                                                             6        11
 + 33*cosh(a)*sinh(a)*sinh(b)*sinh(c) - 16384*cosh(b)*sinh(a) *sinh(c)

                        6        9                        6        7
 - 45056*cosh(b)*sinh(a) *sinh(c)  - 45056*cosh(b)*sinh(a) *sinh(c)

                        6        5                       6        3
 - 19712*cosh(b)*sinh(a) *sinh(c)  - 3520*cosh(b)*sinh(a) *sinh(c)

                      6                                4        11
 - 176*cosh(b)*sinh(a) *sinh(c) - 24576*cosh(b)*sinh(a) *sinh(c)

                        4        9                        4        7
 - 67584*cosh(b)*sinh(a) *sinh(c)  - 67584*cosh(b)*sinh(a) *sinh(c)

                        4        5                       4        3
 - 29568*cosh(b)*sinh(a) *sinh(c)  - 5280*cosh(b)*sinh(a) *sinh(c)

                      4                               2        11
 - 264*cosh(b)*sinh(a) *sinh(c) - 9216*cosh(b)*sinh(a) *sinh(c)

                        2        9                        2        7
 - 25344*cosh(b)*sinh(a) *sinh(c)  - 25344*cosh(b)*sinh(a) *sinh(c)

                        2        5                       2        3
 - 11088*cosh(b)*sinh(a) *sinh(c)  - 1980*cosh(b)*sinh(a) *sinh(c)

                     2                              8
 - 99*cosh(b)*sinh(a) *sinh(c) + 128*cosh(b)*sinh(b) *sinh(c)

                      6                              4
 + 224*cosh(b)*sinh(b) *sinh(c) + 120*cosh(b)*sinh(b) *sinh(c)

                     2                              11                       9
 + 20*cosh(b)*sinh(b) *sinh(c) - 512*cosh(b)*sinh(c)   - 1408*cosh(b)*sinh(c)

                       7                      5                      3
 - 1408*cosh(b)*sinh(c)  - 616*cosh(b)*sinh(c)  - 110*cosh(b)*sinh(c)

                                            6                10
 - 5*cosh(b)*sinh(c) - 16384*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        6                8
 - 36864*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        6                6
 - 28672*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                       6                4
 - 8960*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                      6                2                     6
 - 960*cosh(c)*sinh(a) *sinh(b)*sinh(c)  - 16*cosh(c)*sinh(a) *sinh(b)

                        4                10
 - 24576*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        4                8
 - 55296*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        4                6
 - 43008*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        4                4
 - 13440*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                       4                2                     4
 - 1440*cosh(c)*sinh(a) *sinh(b)*sinh(c)  - 24*cosh(c)*sinh(a) *sinh(b)

                       2                10
 - 9216*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        2                8
 - 20736*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                        2                6
 - 16128*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                       2                4
 - 5040*cosh(c)*sinh(a) *sinh(b)*sinh(c)

                      2                2                    2
 - 540*cosh(c)*sinh(a) *sinh(b)*sinh(c)  - 9*cosh(c)*sinh(a) *sinh(b)

                      9                      7                      5
 + 128*cosh(c)*sinh(b)  + 288*cosh(c)*sinh(b)  + 216*cosh(c)*sinh(b)

                     3                              10
 + 60*cosh(c)*sinh(b)  - 512*cosh(c)*sinh(b)*sinh(c)

                               8                              6
 - 1152*cosh(c)*sinh(b)*sinh(c)  - 896*cosh(c)*sinh(b)*sinh(c)

                              4                             2
 - 280*cosh(c)*sinh(b)*sinh(c)  - 30*cosh(c)*sinh(b)*sinh(c)

 + 4*cosh(c)*sinh(b)

trigsimp(ws, combine);


 sinh(9*b + c) + sinh(6*a - b - 11*c)
--------------------------------------
                  2

trigsimp(sec(20x-y), keepalltrig);


       20              20                20                   19
(csc(x)  *csc(y)*sec(x)  *sec(y))/(csc(x)  *csc(y) + 20*csc(x)  *sec(x)*sec(y)

                18              2              17       3
    - 190*csc(x)  *csc(y)*sec(x)  - 1140*csc(x)  *sec(x) *sec(y)

                 16              4               15       5
    + 4845*csc(x)  *csc(y)*sec(x)  + 15504*csc(x)  *sec(x) *sec(y)

                  14              6               13       7
    - 38760*csc(x)  *csc(y)*sec(x)  - 77520*csc(x)  *sec(x) *sec(y)

                   12              8                11       9
    + 125970*csc(x)  *csc(y)*sec(x)  + 167960*csc(x)  *sec(x) *sec(y)

                   10              10                9       11
    - 184756*csc(x)  *csc(y)*sec(x)   - 167960*csc(x) *sec(x)  *sec(y)

                   8              12               7       13
    + 125970*csc(x) *csc(y)*sec(x)   + 77520*csc(x) *sec(x)  *sec(y)

                  6              14               5       15
    - 38760*csc(x) *csc(y)*sec(x)   - 15504*csc(x) *sec(x)  *sec(y)

                 4              16              3       17
    + 4845*csc(x) *csc(y)*sec(x)   + 1140*csc(x) *sec(x)  *sec(y)

                2              18                   19                       20
    - 190*csc(x) *csc(y)*sec(x)   - 20*csc(x)*sec(x)  *sec(y) + csc(y)*sec(x)  )

trigsimp(csc(10a-9b), keepalltrig);


          10       9       10       9           10       8
( - csc(a)  *csc(b) *sec(a)  *sec(b) )/(9*csc(a)  *csc(b) *sec(b)

               10       6       3             10       4       5
    - 84*csc(a)  *csc(b) *sec(b)  + 126*csc(a)  *csc(b) *sec(b)

               10       2       7         10       9            9       9
    - 36*csc(a)  *csc(b) *sec(b)  + csc(a)  *sec(b)  - 10*csc(a) *csc(b) *sec(a)

                9       7              2              9       5              4
    + 360*csc(a) *csc(b) *sec(a)*sec(b)  - 1260*csc(a) *csc(b) *sec(a)*sec(b)

                9       3              6            9                     8
    + 840*csc(a) *csc(b) *sec(a)*sec(b)  - 90*csc(a) *csc(b)*sec(a)*sec(b)

                8       8       2                     8       6       2       3
    - 405*csc(a) *csc(b) *sec(a) *sec(b) + 3780*csc(a) *csc(b) *sec(a) *sec(b)

                 8       4       2       5
    - 5670*csc(a) *csc(b) *sec(a) *sec(b)

                 8       2       2       7            8       2       9
    + 1620*csc(a) *csc(b) *sec(a) *sec(b)  - 45*csc(a) *sec(a) *sec(b)

                7       9       3              7       7       3       2
    + 120*csc(a) *csc(b) *sec(a)  - 4320*csc(a) *csc(b) *sec(a) *sec(b)

                  7       5       3       4
    + 15120*csc(a) *csc(b) *sec(a) *sec(b)

                  7       3       3       6
    - 10080*csc(a) *csc(b) *sec(a) *sec(b)

                 7              3       8              6       8       4
    + 1080*csc(a) *csc(b)*sec(a) *sec(b)  + 1890*csc(a) *csc(b) *sec(a) *sec(b)

                  6       6       4       3
    - 17640*csc(a) *csc(b) *sec(a) *sec(b)

                  6       4       4       5
    + 26460*csc(a) *csc(b) *sec(a) *sec(b)

                 6       2       4       7             6       4       9
    - 7560*csc(a) *csc(b) *sec(a) *sec(b)  + 210*csc(a) *sec(a) *sec(b)

                5       9       5              5       7       5       2
    - 252*csc(a) *csc(b) *sec(a)  + 9072*csc(a) *csc(b) *sec(a) *sec(b)

                  5       5       5       4
    - 31752*csc(a) *csc(b) *sec(a) *sec(b)

                  5       3       5       6
    + 21168*csc(a) *csc(b) *sec(a) *sec(b)

                 5              5       8              4       8       6
    - 2268*csc(a) *csc(b)*sec(a) *sec(b)  - 1890*csc(a) *csc(b) *sec(a) *sec(b)

                  4       6       6       3
    + 17640*csc(a) *csc(b) *sec(a) *sec(b)

                  4       4       6       5
    - 26460*csc(a) *csc(b) *sec(a) *sec(b)

                 4       2       6       7             4       6       9
    + 7560*csc(a) *csc(b) *sec(a) *sec(b)  - 210*csc(a) *sec(a) *sec(b)

                3       9       7              3       7       7       2
    + 120*csc(a) *csc(b) *sec(a)  - 4320*csc(a) *csc(b) *sec(a) *sec(b)

                  3       5       7       4
    + 15120*csc(a) *csc(b) *sec(a) *sec(b)

                  3       3       7       6
    - 10080*csc(a) *csc(b) *sec(a) *sec(b)

                 3              7       8             2       8       8
    + 1080*csc(a) *csc(b)*sec(a) *sec(b)  + 405*csc(a) *csc(b) *sec(a) *sec(b)

                 2       6       8       3
    - 3780*csc(a) *csc(b) *sec(a) *sec(b)

                 2       4       8       5
    + 5670*csc(a) *csc(b) *sec(a) *sec(b)

                 2       2       8       7            2       8       9
    - 1620*csc(a) *csc(b) *sec(a) *sec(b)  + 45*csc(a) *sec(a) *sec(b)

                      9       9                    7       9       2
    - 10*csc(a)*csc(b) *sec(a)  + 360*csc(a)*csc(b) *sec(a) *sec(b)

                        5       9       4                    3       9       6
    - 1260*csc(a)*csc(b) *sec(a) *sec(b)  + 840*csc(a)*csc(b) *sec(a) *sec(b)

                             9       8           8       10
    - 90*csc(a)*csc(b)*sec(a) *sec(b)  - 9*csc(b) *sec(a)  *sec(b)

               6       10       3             4       10       5
    + 84*csc(b) *sec(a)  *sec(b)  - 126*csc(b) *sec(a)  *sec(b)

               2       10       7         10       9
    + 36*csc(b) *sec(a)  *sec(b)  - sec(a)  *sec(b) )

trigsimp(ws, combine);


        1
-----------------
 sin(10*a - 9*b)

trigsimp(cosh(50*acosh(x))-cos(50*acos(x)));


0

trigsimp(cos(n*acos(x))-cosh(n*acosh(x)), trig);


0

trigsimp((2tan(log(x))*(sec(log(x))^2 - tan(log(x))^2 - 1))/x);


0

trigsimp(sech(10x), keepalltrig);


        10        10          10             8        2              6        4
(csch(x)  *sech(x)  )/(csch(x)   + 45*csch(x) *sech(x)  + 210*csch(x) *sech(x)

                 4        6             2        8          10
    + 210*csch(x) *sech(x)  + 45*csch(x) *sech(x)  + sech(x)  )

trigsimp(ws, combine);


     1
------------
 cosh(10*x)

trigsimp(csch(3x-5y), keepalltrig);


           3        5        3        5            3        4
( - csch(x) *csch(y) *sech(x) *sech(y) )/(5*csch(x) *csch(y) *sech(y)

                3        2        3          3        5
    + 10*csch(x) *csch(y) *sech(y)  + csch(x) *sech(y)

               2        5                     2        3                2
    - 3*csch(x) *csch(y) *sech(x) - 30*csch(x) *csch(y) *sech(x)*sech(y)

                2                        4
    - 15*csch(x) *csch(y)*sech(x)*sech(y)

                        4        2
    + 15*csch(x)*csch(y) *sech(x) *sech(y)

                        2        2        3                    2        5
    + 30*csch(x)*csch(y) *sech(x) *sech(y)  + 3*csch(x)*sech(x) *sech(y)

             5        3             3        3        2
    - csch(y) *sech(x)  - 10*csch(y) *sech(x) *sech(y)

                       3        4
    - 5*csch(y)*sech(x) *sech(y) )

trigsimp(ws, combine);


        1
-----------------
 sinh(3*x - 5*y)

off precise;


trigsimp((sinh(x)+cosh(x))^n+(cosh(x)-sinh(x))^n, expon);


  2*n*x
 e      + 1
------------
     n*x
    e

on precise;


trigsimp(ws, hyp);


2*cosh(n*x)

load_package taylor;


taylor(sin(x+a)*cos(x+b), x, 0, 4);


cos(b)*sin(a) + (cos(a)*cos(b) - sin(a)*sin(b))*x

                                    2
 - (cos(a)*sin(b) + cos(b)*sin(a))*x

    2*( - cos(a)*cos(b) + sin(a)*sin(b))   3
 + --------------------------------------*x
                     3

    cos(a)*sin(b) + cos(b)*sin(a)   4      5
 + -------------------------------*x  + O(x )
                  3

trigsimp(ws, combine);


 sin(a - b) + sin(a + b)                               2    2*cos(a + b)   3
------------------------- + cos(a + b)*x - sin(a + b)*x  - --------------*x
            2                                                    3

    sin(a + b)   4      5
 + ------------*x  + O(x )
        3


%-----------------------TrigFactorize-----------------------

on nopowers;

  % for comparison with version 2.0
trigfactorize(sin(x)**2, x);


{sin(x),sin(x)}

trigfactorize(1+cos(x), x);


{cos(x) + 1}

trigfactorize(1+cos(x), x/2);


        x        x
{2,cos(---),cos(---)}
        2        2

trigfactorize(1+cos(x), x/6);


{2,

           x  2
  - 4*sin(---)  + 1,
           6

           x  2
  - 4*sin(---)  + 1,
           6

      x
 cos(---),
      6

      x
 cos(---)}
      6

trigfactorize(sin(x)*(1-cos(x)), x);


{sin(x)*( - cos(x) + 1)}

trigfactorize(sin(x)*(1-cos(x)), x/2);


{4,

      x
 cos(---),
      2

      x
 sin(---),
      2

      x
 sin(---),
      2

      x
 sin(---)}
      2

trigfactorize(tan(x), x);


{tan(x)}

trigfactorize(sin(x*3), x);


            2
{ - 4*sin(x)  + 3,sin(x)}

trigfactorize(sin(4x)-1, x);


{-1,

                           2
 2*cos(x)*sin(x) + 2*sin(x)  - 1,

                           2
 2*cos(x)*sin(x) + 2*sin(x)  - 1}

trigfactorize(sin(x)**4-1, x);


          2
{-1,sin(x)  + 1,cos(x),cos(x)}

trigfactorize(cos(x)**4-1, x);


       2
{sin(x)  - 2,sin(x),sin(x)}

trigfactorize(sin(x)**10-cos(x)**6, x);


{-1,

              2                  5
 cos(x)*sin(x)  - cos(x) - sin(x) ,

              2                  5
 cos(x)*sin(x)  - cos(x) + sin(x) }

trigfactorize(sin(x)*cos(y), x);


{cos(y),sin(x)}

trigfactorize(sin(2x)*cos(y)**2, y/2);


{2*cos(x)*sin(x),

      y          y
 cos(---) + sin(---),
      2          2

      y          y
 cos(---) + sin(---),
      2          2

      y          y
 cos(---) - sin(---),
      2          2

      y          y
 cos(---) - sin(---)}
      2          2

trigfactorize(sin(y)**4-x**2, y);


       2           2
{sin(y)  - x,sin(y)  + x}

trigfactorize(sin(x), x+1);


***** TrigGCD/Factorize error: last arg must be [number*]variable. 

trigfactorize(sin(x), 2x);


***** TrigGCD/Factorize error: basis not possible. 

trigfactorize(sin(x)*cosh(x), x/2);


{2,

      x
 cos(---),
      2

      x
 sin(---),
      2

       x             x
 cosh(---) - i*sinh(---),
       2             2

       x             x
 cosh(---) + i*sinh(---)}
       2             2

trigfactorize(1+cos(2x)+2cos(x)*cosh(x), x/2);


{4,

      x         x            x         x
 cos(---)*cosh(---) + i*sin(---)*sinh(---),
      2         2            2         2

      x         x            x         x
 cos(---)*cosh(---) - i*sin(---)*sinh(---),
      2         2            2         2

      x          x
 cos(---) + sin(---),
      2          2

      x          x
 cos(---) - sin(---)}
      2          2


%-------------------------TrigGCD---------------------------

triggcd(sin(x), cos(x), x);


1

triggcd(1-cos(x)^2, sin(x)^2, x);


         2
 - sin(x)

triggcd(sin(x)^4-1, cos(x)^2, x);


         2
 - sin(x)  + 1

triggcd(sin(5x+1), cos(x), x);


1

triggcd(1-cos(2x), sin(2x), x);


sin(x)

triggcd(-5+cos(2x)-6sin(x), -7+cos(2x)-8sin(x), x/2);


       x        x
2*cos(---)*sin(---) + 1
       2        2

triggcd(1-2cosh(x)+cosh(2x), 1+2cosh(x)+cosh(2x), x/2);


        x  2
2*sinh(---)  + 1
        2

triggcd(1+cos(2x)+2cos(x)*cosh(x), 1+2cos(x)*cosh(x)+cosh(2x), x/2);


        x  2         x  2
 - sin(---)  + sinh(---)  + 1
        2            2

triggcd(-1+2a*b+cos(2x)-2a*sin(x)+2b*sin(x),
   -1-2a*b+cos(2x)-2a*sin(x)-2b*sin(x), x/2);


       x        x
2*cos(---)*sin(---) + a
       2        2

triggcd(sin(x)^10-1, cos(x), x);


cos(x)

triggcd(sin(5x)+sin(3x), cos(x), x);


cos(x)

triggcd(sin(3x)+sin(5x), sin(5x)+sin(7x), x);


              2
sin(x)*(sin(x)  - 1)


%-----------------------------------------------------------

% New facilities in version 2

%-----------------------------------------------------------

% TrigSimp applied to non-scalars data structures:
trigsimp( sin(2x) = cos(2x) );


                           2
2*cos(x)*sin(x)= - 2*sin(x)  + 1

trigsimp( { sin(2x), cos(2x) } );


                            2
{2*cos(x)*sin(x), - 2*sin(x)  + 1}

trigsimp( { sin(2x) = cos(2x) } );


                            2
{2*cos(x)*sin(x)= - 2*sin(x)  + 1}

trigsimp( mat((sin(2x),cos(2x)),
              (csc(2x),sec(2x))) );


[                              2    ]
[ 2*cos(x)*sin(x)    - 2*sin(x)  + 1]
[                                   ]
[        1                - 1       ]
[-----------------  --------------- ]
[ 2*cos(x)*sin(x)            2      ]
[                    2*sin(x)  - 1  ]



% An amusing identify:
trigsimp(csc x - cot x - tan(x/2));


0


% which could be DERIVED like this:
trigsimp(csc x - cot x, x/2, tan);


     x
tan(---)
     2


% A silly illustration of multiple additional trig arguments:
trigsimp(csc x - cot x, x/2, x/3);


                  x  5           x  3          x
          16*sin(---)  - 24*sin(---)  + 9*sin(---)
                  6              6             6
------------------------------------------------------------
         x        x  4           x        x  2          x
 16*cos(---)*sin(---)  - 16*cos(---)*sin(---)  + 3*cos(---)
         6        6              6        6             6


% A more useful illustration of multiple additional trig arguments:
trigsimp(csc x - cot x + csc y - cot y, x/2, y/2, tan);


     x          y
tan(---) + tan(---)
     2          2


%-----------------------------------------------------------

% New TrigFactorize facility:
off nopowers;

  % REDUCE 3.7 default, gives more compact output ...
trigfactorize(sin(x)^2, x);


{{sin(x),2}}

trigfactorize(1+cos(x), x);


{{cos(x) + 1,1}}

trigfactorize(1+cos(x), x/2);


             x
{{2,1},{cos(---),2}}
             2

trigfactorize(1+cos(x), x/6);


             x                 x  2
{{2,1},{cos(---),2},{ - 4*sin(---)  + 1,2}}
             6                 6

trigfactorize(sin(x)*(1-cos(x)), x);


{{sin(x)*( - cos(x) + 1),1}}

trigfactorize(sin(x)*(1-cos(x)), x/2);


             x            x
{{4,1},{sin(---),3},{cos(---),1}}
             2            2

trigfactorize(tan(x), x);


{{tan(x),1}}

trigfactorize(sin(3x), x);


                        2
{{sin(x),1},{ - 4*sin(x)  + 3,1}}

trigfactorize(sin(4x) - 1, x);


                                   2
{{-1,1},{2*cos(x)*sin(x) + 2*sin(x)  - 1,2}}

trigfactorize(sin(x)^4 - 1, x);


                          2
{{-1,1},{cos(x),2},{sin(x)  + 1,1}}

trigfactorize(cos(x)^4 - 1, x);


                   2
{{sin(x),2},{sin(x)  - 2,1}}

trigfactorize(sin(x)^10 - cos(x)^6, x);


{{-1,1},

               2                  5
 {cos(x)*sin(x)  - cos(x) + sin(x) ,1},

               2                  5
 {cos(x)*sin(x)  - cos(x) - sin(x) ,1}}

trigfactorize(sin(x)*cos(y), x);


{{cos(y),1},{sin(x),1}}

trigfactorize(sin(2x)*cos(y)^2, y/2);


{{2*cos(x)*sin(x),1},

       y          y
 {cos(---) - sin(---),2},
       2          2

       y          y
 {cos(---) + sin(---),2}}
       2          2

trigfactorize(sin(y)^4 - x^2, y);


        2               2
{{sin(y)  + x,1},{sin(y)  - x,1}}

trigfactorize(sin(x), x+1);


***** TrigGCD/Factorize error: last arg must be [number*]variable. 

trigfactorize(sin(x), 2x);


***** TrigGCD/Factorize error: basis not possible. 

trigfactorize(sin(x)*cosh(x), x/2);


{{2,1},

        x             x
 {cosh(---) + i*sinh(---),1},
        2             2

        x             x
 {cosh(---) - i*sinh(---),1},
        2             2

       x
 {sin(---),1},
       2

       x
 {cos(---),1}}
       2

trigfactorize(1 + cos(2x) + 2cos(x)*cosh(x), x/2);


{{4,1},

       x          x
 {cos(---) - sin(---),1},
       2          2

       x          x
 {cos(---) + sin(---),1},
       2          2

       x         x            x         x
 {cos(---)*cosh(---) - i*sin(---)*sinh(---),
       2         2            2         2

  1},

       x         x            x         x
 {cos(---)*cosh(---) + i*sin(---)*sinh(---),
       2         2            2         2

  1}}


end;


Time for test: 3492 ms, plus GC time: 85 ms