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Упрощение выражений/ 2*sin(15*x)*cos(15*x) если x=1/2

2*sin(15*x)*cos(15*x) если x=1/2

Выражение, которое надо упростить:

Решение

Вы ввели [src] 
2*sin(15*x)*cos(15*x)
$$2 \sin{\left(15 x \right)} \cos{\left(15 x \right)}$$ 
2*sin(15*x)*cos(15*x)
Общее упрощение [src] 
sin(30*x)
$$\sin{\left(30 x \right)}$$ 
sin(30*x)
Подстановка условия [src] 
2*sin(15*x)*cos(15*x) при x = 1/2
подставляем
2*sin(15*x)*cos(15*x)
$$2 \sin{\left(15 x \right)} \cos{\left(15 x \right)}$$ 
sin(30*x)
$$\sin{\left(30 x \right)}$$ 
переменные
x = 1/2
$$x = \frac{1}{2}$$ 
sin(30*(1/2))
$$\sin{\left(30 (1/2) \right)}$$ 
sin(15)
$$\sin{\left(15 \right)}$$ 
sin(15)
Численный ответ [src] 
2.0*cos(15*x)*sin(15*x)
2.0*cos(15*x)*sin(15*x)
Степени [src] 
   / -15*I*x    15*I*x\                       
   |e          e      | /   -15*I*x    15*I*x\
-I*|-------- + -------|*\- e        + e      /
   \   2          2   /
$$- i \left(\frac{e^{15 i x}}{2} + \frac{e^{- 15 i x}}{2}\right) \left(e^{15 i x} - e^{- 15 i x}\right)$$ 
-i*(exp(-15*i*x)/2 + exp(15*i*x)/2)*(-exp(-15*i*x) + exp(15*i*x))
Собрать выражение [src] 
sin(30*x)
$$\sin{\left(30 x \right)}$$ 
sin(30*x)
Раскрыть выражение [src] 
/     15              8       7              4       11             12       3            14                    2       13              10       5              6       9   \ /     15               9       6              5       10             13       2            14                    3       12              11       4               7       8   \
\- sin  (x) - 6435*cos (x)*sin (x) - 1365*cos (x)*sin  (x) - 455*cos  (x)*sin (x) + 15*cos  (x)*sin(x) + 105*cos (x)*sin  (x) + 3003*cos  (x)*sin (x) + 5005*cos (x)*sin (x)/*\2*cos  (x) - 10010*cos (x)*sin (x) - 6006*cos (x)*sin  (x) - 210*cos  (x)*sin (x) - 30*sin  (x)*cos(x) + 910*cos (x)*sin  (x) + 2730*cos  (x)*sin (x) + 12870*cos (x)*sin (x)/
$$\left(- 30 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 910 \sin^{12}{\left(x \right)} \cos^{3}{\left(x \right)} - 6006 \sin^{10}{\left(x \right)} \cos^{5}{\left(x \right)} + 12870 \sin^{8}{\left(x \right)} \cos^{7}{\left(x \right)} - 10010 \sin^{6}{\left(x \right)} \cos^{9}{\left(x \right)} + 2730 \sin^{4}{\left(x \right)} \cos^{11}{\left(x \right)} - 210 \sin^{2}{\left(x \right)} \cos^{13}{\left(x \right)} + 2 \cos^{15}{\left(x \right)}\right) \left(- \sin^{15}{\left(x \right)} + 105 \sin^{13}{\left(x \right)} \cos^{2}{\left(x \right)} - 1365 \sin^{11}{\left(x \right)} \cos^{4}{\left(x \right)} + 5005 \sin^{9}{\left(x \right)} \cos^{6}{\left(x \right)} - 6435 \sin^{7}{\left(x \right)} \cos^{8}{\left(x \right)} + 3003 \sin^{5}{\left(x \right)} \cos^{10}{\left(x \right)} - 455 \sin^{3}{\left(x \right)} \cos^{12}{\left(x \right)} + 15 \sin{\left(x \right)} \cos^{14}{\left(x \right)}\right)$$ 
                 11       11                    9       9                    9       13                    13       9                    13       13                    7       11                    11       7                    11       15                    15       11                    7       7                   7       15                   15       7                   5       9                   9       5                   5       13                   13       5                   15       15                   3       11                   11       3                  5       5                  3       7                  7       3                  3       15                  15       3                 9                        9                        13                        13                       3       3                5                       5                                          3                      3                       15                       15                       7                       7                        11                        11                        3       5                 5       3                  3       13                  13       3                  3       9                  9       3                   5       15                   15       5                   5       7                   7       5                    5       11                    11       5                    13       15                    15       13                    9       15                    15       9                    7       13                    13       7                    7       9                    9       7                     11       13                     13       11                     9       11                     11       9   
- 16986931200*cos  (x)*sin  (x) - 9912320000*cos (x)*sin (x) - 8650752000*cos (x)*sin  (x) - 8650752000*cos  (x)*sin (x) - 7549747200*cos  (x)*sin  (x) - 5308416000*cos (x)*sin  (x) - 5308416000*cos  (x)*sin (x) - 3019898880*cos  (x)*sin  (x) - 3019898880*cos  (x)*sin  (x) - 1658880000*cos (x)*sin (x) - 943718400*cos (x)*sin  (x) - 943718400*cos  (x)*sin (x) - 851558400*cos (x)*sin (x) - 851558400*cos (x)*sin (x) - 743178240*cos (x)*sin  (x) - 743178240*cos  (x)*sin (x) - 536870912*cos  (x)*sin  (x) - 103219200*cos (x)*sin  (x) - 103219200*cos  (x)*sin (x) - 73156608*cos (x)*sin (x) - 32256000*cos (x)*sin (x) - 32256000*cos (x)*sin (x) - 18350080*cos (x)*sin  (x) - 18350080*cos  (x)*sin (x) - 2112000*cos (x)*sin(x) - 2112000*sin (x)*cos(x) - 1843200*cos  (x)*sin(x) - 1843200*sin  (x)*cos(x) - 627200*cos (x)*sin (x) - 181440*cos (x)*sin(x) - 181440*sin (x)*cos(x) - 450*cos(x)*sin(x) + 16800*cos (x)*sin(x) + 16800*sin (x)*cos(x) + 491520*cos  (x)*sin(x) + 491520*sin  (x)*cos(x) + 864000*cos (x)*sin(x) + 864000*sin (x)*cos(x) + 2764800*cos  (x)*sin(x) + 2764800*sin  (x)*cos(x) + 6773760*cos (x)*sin (x) + 6773760*cos (x)*sin (x) + 68812800*cos (x)*sin  (x) + 68812800*cos  (x)*sin (x) + 78848000*cos (x)*sin (x) + 78848000*cos (x)*sin (x) + 198180864*cos (x)*sin  (x) + 198180864*cos  (x)*sin (x) + 348364800*cos (x)*sin (x) + 348364800*cos (x)*sin (x) + 1114767360*cos (x)*sin  (x) + 1114767360*cos  (x)*sin (x) + 2013265920*cos  (x)*sin  (x) + 2013265920*cos  (x)*sin  (x) + 2306867200*cos (x)*sin  (x) + 2306867200*cos  (x)*sin (x) + 3538944000*cos (x)*sin  (x) + 3538944000*cos  (x)*sin (x) + 4055040000*cos (x)*sin (x) + 4055040000*cos (x)*sin (x) + 11324620800*cos  (x)*sin  (x) + 11324620800*cos  (x)*sin  (x) + 12976128000*cos (x)*sin  (x) + 12976128000*cos  (x)*sin (x)
$$- 536870912 \sin^{15}{\left(x \right)} \cos^{15}{\left(x \right)} + 2013265920 \sin^{15}{\left(x \right)} \cos^{13}{\left(x \right)} + 2013265920 \sin^{13}{\left(x \right)} \cos^{15}{\left(x \right)} - 3019898880 \sin^{15}{\left(x \right)} \cos^{11}{\left(x \right)} - 7549747200 \sin^{13}{\left(x \right)} \cos^{13}{\left(x \right)} - 3019898880 \sin^{11}{\left(x \right)} \cos^{15}{\left(x \right)} + 2306867200 \sin^{15}{\left(x \right)} \cos^{9}{\left(x \right)} + 11324620800 \sin^{13}{\left(x \right)} \cos^{11}{\left(x \right)} + 11324620800 \sin^{11}{\left(x \right)} \cos^{13}{\left(x \right)} + 2306867200 \sin^{9}{\left(x \right)} \cos^{15}{\left(x \right)} - 943718400 \sin^{15}{\left(x \right)} \cos^{7}{\left(x \right)} - 8650752000 \sin^{13}{\left(x \right)} \cos^{9}{\left(x \right)} - 16986931200 \sin^{11}{\left(x \right)} \cos^{11}{\left(x \right)} - 8650752000 \sin^{9}{\left(x \right)} \cos^{13}{\left(x \right)} - 943718400 \sin^{7}{\left(x \right)} \cos^{15}{\left(x \right)} + 198180864 \sin^{15}{\left(x \right)} \cos^{5}{\left(x \right)} + 3538944000 \sin^{13}{\left(x \right)} \cos^{7}{\left(x \right)} + 12976128000 \sin^{11}{\left(x \right)} \cos^{9}{\left(x \right)} + 12976128000 \sin^{9}{\left(x \right)} \cos^{11}{\left(x \right)} + 3538944000 \sin^{7}{\left(x \right)} \cos^{13}{\left(x \right)} + 198180864 \sin^{5}{\left(x \right)} \cos^{15}{\left(x \right)} - 18350080 \sin^{15}{\left(x \right)} \cos^{3}{\left(x \right)} - 743178240 \sin^{13}{\left(x \right)} \cos^{5}{\left(x \right)} - 5308416000 \sin^{11}{\left(x \right)} \cos^{7}{\left(x \right)} - 9912320000 \sin^{9}{\left(x \right)} \cos^{9}{\left(x \right)} - 5308416000 \sin^{7}{\left(x \right)} \cos^{11}{\left(x \right)} - 743178240 \sin^{5}{\left(x \right)} \cos^{13}{\left(x \right)} - 18350080 \sin^{3}{\left(x \right)} \cos^{15}{\left(x \right)} + 491520 \sin^{15}{\left(x \right)} \cos{\left(x \right)} + 68812800 \sin^{13}{\left(x \right)} \cos^{3}{\left(x \right)} + 1114767360 \sin^{11}{\left(x \right)} \cos^{5}{\left(x \right)} + 4055040000 \sin^{9}{\left(x \right)} \cos^{7}{\left(x \right)} + 4055040000 \sin^{7}{\left(x \right)} \cos^{9}{\left(x \right)} + 1114767360 \sin^{5}{\left(x \right)} \cos^{11}{\left(x \right)} + 68812800 \sin^{3}{\left(x \right)} \cos^{13}{\left(x \right)} + 491520 \sin{\left(x \right)} \cos^{15}{\left(x \right)} - 1843200 \sin^{13}{\left(x \right)} \cos{\left(x \right)} - 103219200 \sin^{11}{\left(x \right)} \cos^{3}{\left(x \right)} - 851558400 \sin^{9}{\left(x \right)} \cos^{5}{\left(x \right)} - 1658880000 \sin^{7}{\left(x \right)} \cos^{7}{\left(x \right)} - 851558400 \sin^{5}{\left(x \right)} \cos^{9}{\left(x \right)} - 103219200 \sin^{3}{\left(x \right)} \cos^{11}{\left(x \right)} - 1843200 \sin{\left(x \right)} \cos^{13}{\left(x \right)} + 2764800 \sin^{11}{\left(x \right)} \cos{\left(x \right)} + 78848000 \sin^{9}{\left(x \right)} \cos^{3}{\left(x \right)} + 348364800 \sin^{7}{\left(x \right)} \cos^{5}{\left(x \right)} + 348364800 \sin^{5}{\left(x \right)} \cos^{7}{\left(x \right)} + 78848000 \sin^{3}{\left(x \right)} \cos^{9}{\left(x \right)} + 2764800 \sin{\left(x \right)} \cos^{11}{\left(x \right)} - 2112000 \sin^{9}{\left(x \right)} \cos{\left(x \right)} - 32256000 \sin^{7}{\left(x \right)} \cos^{3}{\left(x \right)} - 73156608 \sin^{5}{\left(x \right)} \cos^{5}{\left(x \right)} - 32256000 \sin^{3}{\left(x \right)} \cos^{7}{\left(x \right)} - 2112000 \sin{\left(x \right)} \cos^{9}{\left(x \right)} + 864000 \sin^{7}{\left(x \right)} \cos{\left(x \right)} + 6773760 \sin^{5}{\left(x \right)} \cos^{3}{\left(x \right)} + 6773760 \sin^{3}{\left(x \right)} \cos^{5}{\left(x \right)} + 864000 \sin{\left(x \right)} \cos^{7}{\left(x \right)} - 181440 \sin^{5}{\left(x \right)} \cos{\left(x \right)} - 627200 \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} - 181440 \sin{\left(x \right)} \cos^{5}{\left(x \right)} + 16800 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 16800 \sin{\left(x \right)} \cos^{3}{\left(x \right)} - 450 \sin{\left(x \right)} \cos{\left(x \right)}$$ 
-16986931200*cos(x)^11*sin(x)^11 - 9912320000*cos(x)^9*sin(x)^9 - 8650752000*cos(x)^9*sin(x)^13 - 8650752000*cos(x)^13*sin(x)^9 - 7549747200*cos(x)^13*sin(x)^13 - 5308416000*cos(x)^7*sin(x)^11 - 5308416000*cos(x)^11*sin(x)^7 - 3019898880*cos(x)^11*sin(x)^15 - 3019898880*cos(x)^15*sin(x)^11 - 1658880000*cos(x)^7*sin(x)^7 - 943718400*cos(x)^7*sin(x)^15 - 943718400*cos(x)^15*sin(x)^7 - 851558400*cos(x)^5*sin(x)^9 - 851558400*cos(x)^9*sin(x)^5 - 743178240*cos(x)^5*sin(x)^13 - 743178240*cos(x)^13*sin(x)^5 - 536870912*cos(x)^15*sin(x)^15 - 103219200*cos(x)^3*sin(x)^11 - 103219200*cos(x)^11*sin(x)^3 - 73156608*cos(x)^5*sin(x)^5 - 32256000*cos(x)^3*sin(x)^7 - 32256000*cos(x)^7*sin(x)^3 - 18350080*cos(x)^3*sin(x)^15 - 18350080*cos(x)^15*sin(x)^3 - 2112000*cos(x)^9*sin(x) - 2112000*sin(x)^9*cos(x) - 1843200*cos(x)^13*sin(x) - 1843200*sin(x)^13*cos(x) - 627200*cos(x)^3*sin(x)^3 - 181440*cos(x)^5*sin(x) - 181440*sin(x)^5*cos(x) - 450*cos(x)*sin(x) + 16800*cos(x)^3*sin(x) + 16800*sin(x)^3*cos(x) + 491520*cos(x)^15*sin(x) + 491520*sin(x)^15*cos(x) + 864000*cos(x)^7*sin(x) + 864000*sin(x)^7*cos(x) + 2764800*cos(x)^11*sin(x) + 2764800*sin(x)^11*cos(x) + 6773760*cos(x)^3*sin(x)^5 + 6773760*cos(x)^5*sin(x)^3 + 68812800*cos(x)^3*sin(x)^13 + 68812800*cos(x)^13*sin(x)^3 + 78848000*cos(x)^3*sin(x)^9 + 78848000*cos(x)^9*sin(x)^3 + 198180864*cos(x)^5*sin(x)^15 + 198180864*cos(x)^15*sin(x)^5 + 348364800*cos(x)^5*sin(x)^7 + 348364800*cos(x)^7*sin(x)^5 + 1114767360*cos(x)^5*sin(x)^11 + 1114767360*cos(x)^11*sin(x)^5 + 2013265920*cos(x)^13*sin(x)^15 + 2013265920*cos(x)^15*sin(x)^13 + 2306867200*cos(x)^9*sin(x)^15 + 2306867200*cos(x)^15*sin(x)^9 + 3538944000*cos(x)^7*sin(x)^13 + 3538944000*cos(x)^13*sin(x)^7 + 4055040000*cos(x)^7*sin(x)^9 + 4055040000*cos(x)^9*sin(x)^7 + 11324620800*cos(x)^11*sin(x)^13 + 11324620800*cos(x)^13*sin(x)^11 + 12976128000*cos(x)^9*sin(x)^11 + 12976128000*cos(x)^11*sin(x)^9
Тригонометрическая часть [src] 
sin(30*x)
$$\sin{\left(30 x \right)}$$ 
    1    
---------
csc(30*x)
$$\frac{1}{\csc{\left(30 x \right)}}$$ 
   /       pi\
cos|30*x - --|
   \       2 /
$$\cos{\left(30 x - \frac{\pi}{2} \right)}$$ 
      1       
--------------
   /       pi\
sec|30*x - --|
   \       2 /
$$\frac{1}{\sec{\left(30 x - \frac{\pi}{2} \right)}}$$ 
         2         
-------------------
csc(15*x)*sec(15*x)
$$\frac{2}{\csc{\left(15 x \right)} \sec{\left(15 x \right)}}$$ 
 2*tan(15*x)  
--------------
       2      
1 + tan (15*x)
$$\frac{2 \tan{\left(15 x \right)}}{\tan^{2}{\left(15 x \right)} + 1}$$ 
               /       pi\
2*cos(15*x)*cos|15*x - --|
               \       2 /
$$2 \cos{\left(15 x \right)} \cos{\left(15 x - \frac{\pi}{2} \right)}$$ 
               /pi       \
2*sin(15*x)*sin|-- + 15*x|
               \2        /
$$2 \sin{\left(15 x \right)} \sin{\left(15 x + \frac{\pi}{2} \right)}$$ 
           2            
------------------------
             /       pi\
sec(15*x)*sec|15*x - --|
             \       2 /
$$\frac{2}{\sec{\left(15 x \right)} \sec{\left(15 x - \frac{\pi}{2} \right)}}$$ 
           2            
------------------------
             /pi       \
csc(15*x)*csc|-- - 15*x|
             \2        /
$$\frac{2}{\csc{\left(15 x \right)} \csc{\left(- 15 x + \frac{\pi}{2} \right)}}$$ 
           2            
------------------------
             /pi       \
sec(15*x)*sec|-- - 15*x|
             \2        /
$$\frac{2}{\sec{\left(15 x \right)} \sec{\left(- 15 x + \frac{\pi}{2} \right)}}$$ 
              2              
-----------------------------
                  /pi       \
csc(pi - 15*x)*csc|-- - 15*x|
                  \2        /
$$\frac{2}{\csc{\left(- 15 x + \pi \right)} \csc{\left(- 15 x + \frac{\pi}{2} \right)}}$$ 
  /          2/15*x\\          
2*|-1 + 2*cos |----||*sin(15*x)
  \           \ 2  //
$$2 \cdot \left(2 \cos^{2}{\left(\frac{15 x}{2} \right)} - 1\right) \sin{\left(15 x \right)}$$ 
                               /15*x\
2*(1 + cos(15*x))*cos(15*x)*tan|----|
                               \ 2  /
$$2 \left(\cos{\left(15 x \right)} + 1\right) \cos{\left(15 x \right)} \tan{\left(\frac{15 x}{2} \right)}$$ 
/    0      for 30*x mod pi = 0
<                              
\sin(30*x)       otherwise
$$\begin{cases} 0 & \text{for}\: 30 x \bmod \pi = 0 \\\sin{\left(30 x \right)} & \text{otherwise} \end{cases}$$ 
  /          2/15*x\\    /15*x\    /15*x\
4*|-1 + 2*cos |----||*cos|----|*sin|----|
  \           \ 2  //    \ 2  /    \ 2  /
$$4 \cdot \left(2 \cos^{2}{\left(\frac{15 x}{2} \right)} - 1\right) \sin{\left(\frac{15 x}{2} \right)} \cos{\left(\frac{15 x}{2} \right)}$$ 
/      0         for 30*x mod pi = 0
|                                   
| 2*cot(15*x)                       
<--------------       otherwise     
|       2                           
|1 + cot (15*x)                     
\
$$\begin{cases} 0 & \text{for}\: 30 x \bmod \pi = 0 \\\frac{2 \cot{\left(15 x \right)}}{\cot^{2}{\left(15 x \right)} + 1} & \text{otherwise} \end{cases}$$ 
  /       2/15*x\\    /15*x\
4*|1 - tan |----||*tan|----|
  \        \ 2  //    \ 2  /
----------------------------
                     2      
     /       2/15*x\\       
     |1 + tan |----||       
     \        \ 2  //
$$\frac{4 \cdot \left(- \tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right) \tan{\left(\frac{15 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right)^{2}}$$ 
       /        1     \    
     4*|1 - ----------|    
       |       2/15*x\|    
       |    cot |----||    
       \        \ 2  //    
---------------------------
                2          
/        1     \     /15*x\
|1 + ----------| *cot|----|
|       2/15*x\|     \ 2  /
|    cot |----||           
\        \ 2  //
$$\frac{4 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{15 x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{15 x}{2} \right)}}\right)^{2} \cot{\left(\frac{15 x}{2} \right)}}$$ 
   2/15*x\ /       2/pi   15*x\\ /       2/15*x\\                
cos |----|*|1 - cot |-- + ----||*|1 - tan |----||*(1 + sin(15*x))
    \ 2  / \        \4     2  // \        \ 2  //
$$\left(- \tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(15 x \right)} + 1\right) \cos^{2}{\left(\frac{15 x}{2} \right)}$$ 
           /15*x\    /pi   15*x\      
      8*cot|----|*tan|-- + ----|      
           \ 2  /    \4     2  /      
--------------------------------------
/       2/15*x\\ /       2/pi   15*x\\
|1 + cot |----||*|1 + tan |-- + ----||
\        \ 2  // \        \4     2  //
$$\frac{8 \tan{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{15 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{15 x}{2} \right)} + 1\right)}$$ 
           /15*x\    /pi   15*x\      
      8*tan|----|*tan|-- + ----|      
           \ 2  /    \4     2  /      
--------------------------------------
/       2/15*x\\ /       2/pi   15*x\\
|1 + tan |----||*|1 + tan |-- + ----||
\        \ 2  // \        \4     2  //
$$\frac{8 \tan{\left(\frac{15 x}{2} \right)} \tan{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$ 
  //    0      for 15*x mod pi = 0\ //    1      for 15*x mod 2*pi = 0\
2*|<                              |*|<                                |
  \\sin(15*x)       otherwise     / \\cos(15*x)        otherwise      /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases}\right)$$ 
     2/15*x\ /   2              4/15*x\\          
8*sin |----|*|sin (15*x) - 4*sin |----||*sin(15*x)
      \ 2  / \                   \ 2  //          
--------------------------------------------------
                                      2           
           /   2              4/15*x\\            
           |sin (15*x) + 4*sin |----||            
           \                   \ 2  //
$$\frac{8 \left(- 4 \sin^{4}{\left(\frac{15 x}{2} \right)} + \sin^{2}{\left(15 x \right)}\right) \sin^{2}{\left(\frac{15 x}{2} \right)} \sin{\left(15 x \right)}}{\left(4 \sin^{4}{\left(\frac{15 x}{2} \right)} + \sin^{2}{\left(15 x \right)}\right)^{2}}$$ 
  //      0         for 15*x mod pi = 0\                                    
  ||                                   | //    1      for 15*x mod 2*pi = 0\
2*|<   /       pi\                     |*|<                                |
  ||cos|15*x - --|       otherwise     | \\cos(15*x)        otherwise      /
  \\   \       2 /                     /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\cos{\left(15 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases}\right)$$ 
                                    //      1         for 15*x mod 2*pi = 0\
  //    0      for 15*x mod pi = 0\ ||                                     |
2*|<                              |*|<   /pi       \                       |
  \\sin(15*x)       otherwise     / ||sin|-- + 15*x|        otherwise      |
                                    \\   \2        /                       /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\sin{\left(15 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$ 
  /        2/15*x\\ /        2/pi   15*x\\
2*|-1 + cot |----||*|-1 + tan |-- + ----||
  \         \ 2  // \         \4     2  //
------------------------------------------
  /       2/15*x\\ /       2/pi   15*x\\  
  |1 + cot |----||*|1 + tan |-- + ----||  
  \        \ 2  // \        \4     2  //
$$\frac{2 \left(\tan^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{15 x}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{15 x}{2} \right)} + 1\right)}$$ 
             /         4/15*x\\
             |    4*sin |----||
     2/15*x\ |          \ 2  /|
8*sin |----|*|1 - ------------|
      \ 2  / |        2       |
             \     sin (15*x) /
-------------------------------
                   2           
 /         4/15*x\\            
 |    4*sin |----||            
 |          \ 2  /|            
 |1 + ------------| *sin(15*x) 
 |        2       |            
 \     sin (15*x) /
$$\frac{8 \left(- \frac{4 \sin^{4}{\left(\frac{15 x}{2} \right)}}{\sin^{2}{\left(15 x \right)}} + 1\right) \sin^{2}{\left(\frac{15 x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{15 x}{2} \right)}}{\sin^{2}{\left(15 x \right)}} + 1\right)^{2} \sin{\left(15 x \right)}}$$ 
  /       2/pi   15*x\\ /       2/15*x\\
2*|1 - cot |-- + ----||*|1 - tan |----||
  \        \4     2  // \        \ 2  //
----------------------------------------
 /       2/pi   15*x\\ /       2/15*x\\ 
 |1 + cot |-- + ----||*|1 + tan |----|| 
 \        \4     2  // \        \ 2  //
$$\frac{2 \cdot \left(- \tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$ 
                                      //               /       3*pi\             \
  //    1      for 15*x mod 2*pi = 0\ ||    1      for |15*x + ----| mod 2*pi = 0|
2*|<                                |*|<               \        2  /             |
  \\cos(15*x)        otherwise      / ||                                         |
                                      \\sin(15*x)            otherwise           /
$$2 \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(15 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases}\right)$$ 
  //      0         for 15*x mod pi = 0\                                    
  ||                                   | //    1      for 15*x mod 2*pi = 0\
  ||      1                            | ||                                |
2*|<--------------       otherwise     |*|<    1                           |
  ||   /       pi\                     | ||---------        otherwise      |
  ||sec|15*x - --|                     | \\sec(15*x)                       /
  \\   \       2 /                     /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{1}{\sec{\left(15 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(15 x \right)}} & \text{otherwise} \end{cases}\right)$$ 
                                    //      1         for 15*x mod 2*pi = 0\
  //    0      for 15*x mod pi = 0\ ||                                     |
  ||                              | ||      1                              |
2*|<    1                         |*|<--------------        otherwise      |
  ||---------       otherwise     | ||   /pi       \                       |
  \\csc(15*x)                     / ||csc|-- - 15*x|                       |
                                    \\   \2        /                       /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{1}{\csc{\left(15 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 15 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$ 
  //      0        for 15*x mod pi = 0\                                    
  ||                                  |                                    
  ||1 - cos(15*x)                     | //    1      for 15*x mod 2*pi = 0\
2*|<-------------       otherwise     |*|<                                |
  ||     /15*x\                       | \\cos(15*x)        otherwise      /
  ||  tan|----|                       |                                    
  \\     \ 2  /                       /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{- \cos{\left(15 x \right)} + 1}{\tan{\left(\frac{15 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases}\right)$$ 
     /           2/15*x\   \             
     |        sec |----|   |             
     |            \ 2  /   |    /15*x\   
   4*|1 - -----------------|*sec|----|   
     |       2/  pi   15*x\|    \ 2  /   
     |    sec |- -- + ----||             
     \        \  2     2  //             
-----------------------------------------
                       2                 
/           2/15*x\   \                  
|        sec |----|   |                  
|            \ 2  /   |     /  pi   15*x\
|1 + -----------------| *sec|- -- + ----|
|       2/  pi   15*x\|     \  2     2  /
|    sec |- -- + ----||                  
\        \  2     2  //
$$\frac{4 \left(- \frac{\sec^{2}{\left(\frac{15 x}{2} \right)}}{\sec^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{15 x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{15 x}{2} \right)}}{\sec^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}$$ 
                                    //                                    /pi       \           \
                                    ||              0                 for |-- + 15*x| mod pi = 0|
  //    0      for 15*x mod pi = 0\ ||                                    \2        /           |
2*|<                              |*|<                                                          |
  \\sin(15*x)       otherwise     / ||                   /pi   15*x\                            |
                                    ||(1 + sin(15*x))*cot|-- + ----|          otherwise         |
                                    \\                   \4     2  /                            /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(15 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(15 x \right)} + 1\right) \cot{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$ 
  /       2/  pi   15*x\\                 
  |    cos |- -- + ----||                 
  |        \  2     2  /|    /  pi   15*x\
4*|1 - -----------------|*cos|- -- + ----|
  |           2/15*x\   |    \  2     2  /
  |        cos |----|   |                 
  \            \ 2  /   /                 
------------------------------------------
                           2              
    /       2/  pi   15*x\\               
    |    cos |- -- + ----||               
    |        \  2     2  /|     /15*x\    
    |1 + -----------------| *cos|----|    
    |           2/15*x\   |     \ 2  /    
    |        cos |----|   |               
    \            \ 2  /   /
$$\frac{4 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{15 x}{2} \right)}}\right) \cos{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{15 x}{2} \right)}}\right)^{2} \cos{\left(\frac{15 x}{2} \right)}}$$ 
  /       2/pi   15*x\\               
  |    csc |-- - ----||               
  |        \2     2  /|    /pi   15*x\
4*|1 - ---------------|*csc|-- - ----|
  |          2/15*x\  |    \2     2  /
  |       csc |----|  |               
  \           \ 2  /  /               
--------------------------------------
                        2             
   /       2/pi   15*x\\              
   |    csc |-- - ----||              
   |        \2     2  /|     /15*x\   
   |1 + ---------------| *csc|----|   
   |          2/15*x\  |     \ 2  /   
   |       csc |----|  |              
   \           \ 2  /  /
$$\frac{4 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{15 x}{2} \right)}}\right) \csc{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{15 x}{2} \right)}}\right)^{2} \csc{\left(\frac{15 x}{2} \right)}}$$ 
  //      0         for 15*x mod pi = 0\ //       1         for 15*x mod 2*pi = 0\
  ||                                   | ||                                      |
  ||      /15*x\                       | ||        2/15*x\                       |
  || 2*cot|----|                       | ||-1 + cot |----|                       |
2*|<      \ 2  /                       |*|<         \ 2  /                       |
  ||--------------       otherwise     | ||---------------        otherwise      |
  ||       2/15*x\                     | ||        2/15*x\                       |
  ||1 + cot |----|                     | || 1 + cot |----|                       |
  \\        \ 2  /                     / \\         \ 2  /                       /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{15 x}{2} \right)}}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{15 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$ 
  //      0         for 15*x mod pi = 0\ //      1         for 15*x mod 2*pi = 0\
  ||                                   | ||                                     |
  ||      /15*x\                       | ||       2/15*x\                       |
  || 2*tan|----|                       | ||1 - tan |----|                       |
2*|<      \ 2  /                       |*|<        \ 2  /                       |
  ||--------------       otherwise     | ||--------------        otherwise      |
  ||       2/15*x\                     | ||       2/15*x\                       |
  ||1 + tan |----|                     | ||1 + tan |----|                       |
  \\        \ 2  /                     / \\        \ 2  /                       /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{15 x}{2} \right)}}{\tan^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{15 x}{2} \right)} + 1}{\tan^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$ 
  //               0                 for 15*x mod pi = 0\ //                1                  for 15*x mod 2*pi = 0\
  ||                                                    | ||                                                        |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$ 
                                                     //       1         for 15*x mod 2*pi = 0\
                                                     ||                                      |
  //            0               for 15*x mod pi = 0\ ||         1                            |
  ||                                               | ||-1 + ----------                       |
  ||            2                                  | ||        2/15*x\                       |
  ||--------------------------       otherwise     | ||     tan |----|                       |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{15 x}{2} \right)}}\right) \tan{\left(\frac{15 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{15 x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{15 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$ 
                                         //                         /pi       \           \
  //      0         for 15*x mod pi = 0\ ||         0           for |-- + 15*x| mod pi = 0|
  ||                                   | ||                         \2        /           |
  ||      /15*x\                       | ||                                               |
  || 2*cot|----|                       | ||       /pi   15*x\                             |
2*|<      \ 2  /                       |*|<  2*cot|-- + ----|                             |
  ||--------------       otherwise     | ||       \4     2  /                             |
  ||       2/15*x\                     | ||-------------------          otherwise         |
  ||1 + cot |----|                     | ||       2/pi   15*x\                            |
  \\        \ 2  /                     / ||1 + cot |-- + ----|                            |
                                         \\        \4     2  /                            /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{15 x}{2} \right)}}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(15 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$ 
                                            //                          /       3*pi\             \
  //       1         for 15*x mod 2*pi = 0\ ||         1            for |15*x + ----| mod 2*pi = 0|
  ||                                      | ||                          \        2  /             |
  ||        2/15*x\                       | ||                                                    |
  ||-1 + cot |----|                       | ||        2/pi   15*x\                                |
2*|<         \ 2  /                       |*|<-1 + tan |-- + ----|                                |
  ||---------------        otherwise      | ||         \4     2  /                                |
  ||        2/15*x\                       | ||--------------------            otherwise           |
  || 1 + cot |----|                       | ||       2/pi   15*x\                                 |
  \\         \ 2  /                       / ||1 + tan |-- + ----|                                 |
                                            \\        \4     2  /                                 /
$$2 \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{15 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(15 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{15 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$ 
  //                0                   for 15*x mod pi = 0\ //                1                   for 15*x mod 2*pi = 0\
  ||                                                       | ||                                                         |
  ||    -2*sin(30*x) + 4*sin(15*x)                         | ||  -2 - 2*cos(30*x) + 4*cos(15*x)                         |
2*|<----------------------------------       otherwise     |*|<----------------------------------        otherwise      |
  ||                                 2                     | ||                                 2                       |
  ||1 - cos(30*x) + 2*(1 - cos(15*x))                      | ||1 - cos(30*x) + 2*(1 - cos(15*x))                        |
  \\                                                       / \\                                                         /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{4 \sin{\left(15 x \right)} - 2 \sin{\left(30 x \right)}}{2 \left(- \cos{\left(15 x \right)} + 1\right)^{2} - \cos{\left(30 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{4 \cos{\left(15 x \right)} - 2 \cos{\left(30 x \right)} - 2}{2 \left(- \cos{\left(15 x \right)} + 1\right)^{2} - \cos{\left(30 x \right)} + 1} & \text{otherwise} \end{cases}\right)$$ 
                                                        //        1          for 15*x mod 2*pi = 0\
                                                        ||                                        |
  //              0                for 15*x mod pi = 0\ ||         2                              |
  ||                                                  | ||      sin (15*x)                        |
  ||          sin(15*x)                               | ||-1 + ------------                       |
  ||-----------------------------       otherwise     | ||          4/15*x\                       |
  ||/        2       \                                | ||     4*sin |----|                       |
2*|<|     sin (15*x) |    2/15*x\                     |*|<           \ 2  /                       |
  |||1 + ------------|*sin |----|                     | ||-----------------        otherwise      |
  |||         4/15*x\|     \ 2  /                     | ||         2                              |
  |||    4*sin |----||                                | ||      sin (15*x)                        |
  ||\          \ 2  //                                | || 1 + ------------                       |
  \\                                                  / ||          4/15*x\                       |
                                                        ||     4*sin |----|                       |
                                                        \\           \ 2  /                       /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{\sin{\left(15 x \right)}}{\left(1 + \frac{\sin^{2}{\left(15 x \right)}}{4 \sin^{4}{\left(\frac{15 x}{2} \right)}}\right) \sin^{2}{\left(\frac{15 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(15 x \right)}}{4 \sin^{4}{\left(\frac{15 x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(15 x \right)}}{4 \sin^{4}{\left(\frac{15 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$ 
  //                 0                    for 15*x mod pi = 0\ //                   1                     for 15*x mod 2*pi = 0\
  ||                                                         | ||                                                              |
  ||/      0         for 15*x mod pi = 0                     | ||/       1         for 15*x mod 2*pi = 0                       |
  |||                                                        | |||                                                             |
  |||      /15*x\                                            | |||        2/15*x\                                              |
2*|<| 2*cot|----|                                            |*|<|-1 + cot |----|                                              |
  ||<      \ 2  /                              otherwise     | ||<         \ 2  /                               otherwise      |
  |||--------------       otherwise                          | |||---------------        otherwise                             |
  |||       2/15*x\                                          | |||        2/15*x\                                              |
  |||1 + cot |----|                                          | ||| 1 + cot |----|                                              |
  \\\        \ 2  /                                          / \\\         \ 2  /                                              /
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{15 x}{2} \right)}}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{15 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$ 
                                                                   //          1             for 15*x mod 2*pi = 0\
                                                                   ||                                             |
  //                   0                      for 15*x mod pi = 0\ ||            2/15*x\                          |
  ||                                                             | ||         cos |----|                          |
  ||                   /15*x\                                    | ||             \ 2  /                          |
  ||              2*cos|----|                                    | ||-1 + -----------------                       |
  ||                   \ 2  /                                    | ||        2/  pi   15*x\                       |
  ||----------------------------------------       otherwise     | ||     cos |- -- + ----|                       |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{15 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{15 x}{2} \right)}}{\cos^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{15 x}{2} \right)}}{\cos^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{15 x}{2} \right)}}{\cos^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$ 
                                                            //          1             for 15*x mod 2*pi = 0\
                                                            ||                                             |
  //                0                  for 15*x mod pi = 0\ ||        2/  pi   15*x\                       |
  ||                                                      | ||     sec |- -- + ----|                       |
  ||             /  pi   15*x\                            | ||         \  2     2  /                       |
  ||        2*sec|- -- + ----|                            | ||-1 + -----------------                       |
  ||             \  2     2  /                            | ||            2/15*x\                          |
  ||---------------------------------       otherwise     | ||         sec |----|                          |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{15 x}{2} \right)}}\right) \sec{\left(\frac{15 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{15 x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{15 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{15 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$ 
                                                               //         1            for 15*x mod 2*pi = 0\
                                                               ||                                           |
  //                 0                    for 15*x mod pi = 0\ ||           2/15*x\                         |
  ||                                                         | ||        csc |----|                         |
  ||                 /15*x\                                  | ||            \ 2  /                         |
  ||            2*csc|----|                                  | ||-1 + ---------------                       |
  ||                 \ 2  /                                  | ||        2/pi   15*x\                       |
  ||------------------------------------       otherwise     | ||     csc |-- - ----|                       |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 15 x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{15 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{15 x}{2} \right)}}{\csc^{2}{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 15 x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{15 x}{2} \right)}}{\csc^{2}{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{15 x}{2} \right)}}{\csc^{2}{\left(- \frac{15 x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$ 
2*Piecewise((0, Mod(15*x = pi, 0)), (2*csc(15*x/2)/((1 + csc(15*x/2)^2/csc(pi/2 - 15*x/2)^2)*csc(pi/2 - 15*x/2)), True))*Piecewise((1, Mod(15*x = 2*pi, 0)), ((-1 + csc(15*x/2)^2/csc(pi/2 - 15*x/2)^2)/(1 + csc(15*x/2)^2/csc(pi/2 - 15*x/2)^2), True))
    © Господин Экзамен – Калькулятор