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Simplificación de expresiones/ Descomposición en fracciones simples/ ctg^2(7π/9)-1/(sin^2(7π/9))

¿Cómo vas a descomponer esta ctg^2(7π/9)-1/(sin^2(7π/9)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   2/7*pi\       1     
cot |----| - ----------
    \ 9  /      2/7*pi\
             sin |----|
                 \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{7 \pi}{9} \right)}} + \cot^{2}{\left(\frac{7 \pi}{9} \right)}$$ 
cot((7*pi)/9)^2 - 1/sin((7*pi)/9)^2
Simplificación general [src] 
        /pi\
-1 + sin|--|
        \18/
------------
     2/2*pi\
2*sin |----|
      \ 9  /
$$\frac{-1 + \sin{\left(\frac{\pi}{18} \right)}}{2 \sin^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
(-1 + sin(pi/18))/(2*sin(2*pi/9)^2)
Respuesta numérica [src] 
-1.00000000000000
-1.00000000000000
Denominador racional [src] 
        2/2*pi\    2/2*pi\
-1 + cot |----|*sin |----|
         \ 9  /     \ 9  /
--------------------------
           2/2*pi\        
        sin |----|        
            \ 9  /
$$\frac{-1 + \sin^{2}{\left(\frac{2 \pi}{9} \right)} \cot^{2}{\left(\frac{2 \pi}{9} \right)}}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
(-1 + cot(2*pi/9)^2*sin(2*pi/9)^2)/sin(2*pi/9)^2
Denominador común [src] 
   2/2*pi\       1     
cot |----| - ----------
    \ 9  /      2/2*pi\
             sin |----|
                 \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
cot(2*pi/9)^2 - 1/sin(2*pi/9)^2
Potencias [src] 
   2/2*pi\              4           
cot |----| + -----------------------
    \ 9  /                         2
             /   -7*pi*I    7*pi*I\ 
             |   -------    ------| 
             |      9         9   | 
             \- e        + e      /
$$\cot^{2}{\left(\frac{2 \pi}{9} \right)} + \frac{4}{\left(e^{\frac{7 i \pi}{9}} - e^{- \frac{7 i \pi}{9}}\right)^{2}}$$ 
   2/2*pi\       1     
cot |----| - ----------
    \ 9  /      2/2*pi\
             sin |----|
                 \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
cot(2*pi/9)^2 - 1/sin(2*pi/9)^2
Combinatoria [src] 
/       /2*pi\    /2*pi\\ /        /2*pi\    /2*pi\\
|1 + cot|----|*sin|----||*|-1 + cot|----|*sin|----||
\       \ 9  /    \ 9  // \        \ 9  /    \ 9  //
----------------------------------------------------
                        2/2*pi\                     
                     sin |----|                     
                         \ 9  /
$$\frac{\left(-1 + \sin{\left(\frac{2 \pi}{9} \right)} \cot{\left(\frac{2 \pi}{9} \right)}\right) \left(\sin{\left(\frac{2 \pi}{9} \right)} \cot{\left(\frac{2 \pi}{9} \right)} + 1\right)}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
(1 + cot(2*pi/9)*sin(2*pi/9))*(-1 + cot(2*pi/9)*sin(2*pi/9))/sin(2*pi/9)^2
Abrimos la expresión [src] 
   2/2*pi\         1      
cot |----| - -------------
    \ 9  /          /4*pi\
                 cos|----|
             1      \ 9  /
             - - ---------
             2       2
$$- \frac{1}{\frac{1}{2} - \frac{\cos{\left(\frac{4 \pi}{9} \right)}}{2}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
cot(2*pi/9)^2 - 1/(1/2 - cos(4*pi/9)/2)
Unión de expresiones racionales [src] 
        2/2*pi\    2/2*pi\
-1 + cot |----|*sin |----|
         \ 9  /     \ 9  /
--------------------------
           2/2*pi\        
        sin |----|        
            \ 9  /
$$\frac{-1 + \sin^{2}{\left(\frac{2 \pi}{9} \right)} \cot^{2}{\left(\frac{2 \pi}{9} \right)}}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
(-1 + cot(2*pi/9)^2*sin(2*pi/9)^2)/sin(2*pi/9)^2
Parte trigonométrica [src] 
    1            1     
---------- - ----------
   2/2*pi\      2/5*pi\
tan |----|   cos |----|
    \ 9  /       \ 18 /
$$- \frac{1}{\cos^{2}{\left(\frac{5 \pi}{18} \right)}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                   2/pi\    4/-5*pi\
                cos |--|*sec |-----|
     2/-5*pi\       \18/     \  18 /
- sec |-----| + --------------------
      \  18 /            4
$$- \sec^{2}{\left(- \frac{5 \pi}{18} \right)} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)} \sec^{4}{\left(- \frac{5 \pi}{18} \right)}}{4}$$ 
                             2 
             /       2/5*pi\\  
             |1 + cot |----||  
   2/2*pi\   \        \ 36 //  
cot |----| - ------------------
    \ 9  /                    2
             /        2/5*pi\\ 
             |-1 + cot |----|| 
             \         \ 36 //
$$- \frac{\left(1 + \cot^{2}{\left(\frac{5 \pi}{36} \right)}\right)^{2}}{\left(-1 + \cot^{2}{\left(\frac{5 \pi}{36} \right)}\right)^{2}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
                   2/2*pi\
                cos |----|
      1             \ 9  /
- ---------- + -----------
     2/5*pi\      2/-5*pi\
  cos |----|   cos |-----|
      \ 18 /       \  18 /
$$- \frac{1}{\cos^{2}{\left(\frac{5 \pi}{18} \right)}} + \frac{\cos^{2}{\left(\frac{2 \pi}{9} \right)}}{\cos^{2}{\left(- \frac{5 \pi}{18} \right)}}$$ 
                     2/-5*pi\
                  sec |-----|
        2             \  18 /
- ------------- + -----------
          1           2/2*pi\
  1 - ---------    sec |----|
         /4*pi\        \ 9  /
      sec|----|              
         \ 9  /
$$- \frac{2}{1 - \frac{1}{\sec{\left(\frac{4 \pi}{9} \right)}}} + \frac{\sec^{2}{\left(- \frac{5 \pi}{18} \right)}}{\sec^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
    1                2         
---------- - ------------------
   2/2*pi\              2/2*pi\
tan |----|       1 - tan |----|
    \ 9  /               \ 9  /
             1 - --------------
                        2/2*pi\
                 1 + tan |----|
                         \ 9  /
$$- \frac{2}{- \frac{1 - \tan^{2}{\left(\frac{2 \pi}{9} \right)}}{\tan^{2}{\left(\frac{2 \pi}{9} \right)} + 1} + 1} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                    2/pi\  
                 cos |--|  
      1              \18/  
- ---------- + ------------
     2/7*pi\        4/2*pi\
  sin |----|   4*sin |----|
      \ 9  /         \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{7 \pi}{9} \right)}} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
                      2/pi\  
                   cos |--|  
       1               \18/  
- ----------- + -------------
     2/-5*pi\        4/-5*pi\
  cos |-----|   4*cos |-----|
      \  18 /         \  18 /
$$- \frac{1}{\cos^{2}{\left(- \frac{5 \pi}{18} \right)}} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)}}{4 \cos^{4}{\left(- \frac{5 \pi}{18} \right)}}$$ 
          2/2*pi\         2/pi\        2/-7*pi\
       cos |----|      cos |--|     cos |-----|
  1        \ 9  /          \9 /         \  18 /
- - + ----------- - ------------- - -----------
  2      2/-5*pi\        2/-7*pi\         2/pi\
      cos |-----|   4*cos |-----|    4*cos |--|
          \  18 /         \  18 /          \9 /
$$- \frac{\cos^{2}{\left(\frac{\pi}{9} \right)}}{4 \cos^{2}{\left(- \frac{7 \pi}{18} \right)}} - \frac{1}{2} - \frac{\cos^{2}{\left(- \frac{7 \pi}{18} \right)}}{4 \cos^{2}{\left(\frac{\pi}{9} \right)}} + \frac{\cos^{2}{\left(\frac{2 \pi}{9} \right)}}{\cos^{2}{\left(- \frac{5 \pi}{18} \right)}}$$ 
                      2/pi\      2/pi\
                   cot |--|   tan |--|
  1      2/2*pi\       \9 /       \9 /
- - + cot |----| - -------- - --------
  2       \ 9  /      4          4
$$- \frac{\cot^{2}{\left(\frac{\pi}{9} \right)}}{4} - \frac{1}{2} - \frac{\tan^{2}{\left(\frac{\pi}{9} \right)}}{4} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
                2                 4               2
  /       2/pi\\    /       2/pi\\  /       2/pi\\ 
  |1 + tan |--||    |1 + tan |--|| *|1 - tan |--|| 
  \        \9 //    \        \9 //  \        \36// 
- --------------- + -------------------------------
          2/pi\                        2           
     4*tan |--|          /       2/pi\\     4/pi\  
           \9 /       64*|1 + tan |--|| *tan |--|  
                         \        \36//      \9 /
$$- \frac{\left(\tan^{2}{\left(\frac{\pi}{9} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{\pi}{9} \right)}} + \frac{\left(1 - \tan^{2}{\left(\frac{\pi}{36} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{\pi}{9} \right)} + 1\right)^{4}}{64 \left(\tan^{2}{\left(\frac{\pi}{36} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{\pi}{9} \right)}}$$ 
                           2
             /       2/pi\\ 
             |1 + cot |--|| 
   2/2*pi\   \        \9 // 
cot |----| - ---------------
    \ 9  /           2/pi\  
                4*cot |--|  
                      \9 /
$$- \frac{\left(1 + \cot^{2}{\left(\frac{\pi}{9} \right)}\right)^{2}}{4 \cot^{2}{\left(\frac{\pi}{9} \right)}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
    1            1     
---------- - ----------
   2/2*pi\      2/2*pi\
tan |----|   sin |----|
    \ 9  /       \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                   4/2*pi\ 
                csc |----| 
     2/2*pi\        \ 9  / 
- csc |----| + ------------
      \ 9  /        2/4*pi\
               4*csc |----|
                     \ 9  /
$$- \csc^{2}{\left(\frac{2 \pi}{9} \right)} + \frac{\csc^{4}{\left(\frac{2 \pi}{9} \right)}}{4 \csc^{2}{\left(\frac{4 \pi}{9} \right)}}$$ 
                    2/2*pi\
                 cos |----|
       1             \ 9  /
- ----------- + -----------
     2/-5*pi\      2/-5*pi\
  cos |-----|   cos |-----|
      \  18 /       \  18 /
$$- \frac{1}{\cos^{2}{\left(- \frac{5 \pi}{18} \right)}} + \frac{\cos^{2}{\left(\frac{2 \pi}{9} \right)}}{\cos^{2}{\left(- \frac{5 \pi}{18} \right)}}$$ 
    1           2/2*pi\
---------- - csc |----|
   2/2*pi\       \ 9  /
tan |----|             
    \ 9  /
$$- \csc^{2}{\left(\frac{2 \pi}{9} \right)} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                                   2/pi\
                                cot |--|
  1      2/2*pi\       1            \9 /
- - + cot |----| - ---------- - --------
  2       \ 9  /        2/pi\      4    
                   4*cot |--|           
                         \9 /
$$- \frac{\cot^{2}{\left(\frac{\pi}{9} \right)}}{4} - \frac{1}{2} - \frac{1}{4 \cot^{2}{\left(\frac{\pi}{9} \right)}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
-1
$$-1$$ 
    1              2      
---------- - -------------
   2/2*pi\          /4*pi\
tan |----|   1 - cos|----|
    \ 9  /          \ 9  /
$$- \frac{2}{1 - \cos{\left(\frac{4 \pi}{9} \right)}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                             2
             /       2/5*pi\\ 
             |1 + tan |----|| 
    1        \        \ 36 // 
---------- - -----------------
   2/2*pi\                   2
tan |----|   /       2/5*pi\\ 
    \ 9  /   |1 - tan |----|| 
             \        \ 36 //
$$- \frac{\left(\tan^{2}{\left(\frac{5 \pi}{36} \right)} + 1\right)^{2}}{\left(1 - \tan^{2}{\left(\frac{5 \pi}{36} \right)}\right)^{2}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                   2/2*pi\
                csc |----|
       2            \ 9  /
- ----------- + ----------
         1         2/5*pi\
  1 - -------   csc |----|
         /pi\       \ 18 /
      csc|--|             
         \18/
$$- \frac{2}{1 - \frac{1}{\csc{\left(\frac{\pi}{18} \right)}}} + \frac{\csc^{2}{\left(\frac{2 \pi}{9} \right)}}{\csc^{2}{\left(\frac{5 \pi}{18} \right)}}$$ 
                                 2                  
                  /        2/pi\\     4/pi\    4/pi\
                  |-1 + cot |--|| *sin |--|*tan |--|
        2         \         \36//      \36/     \9 /
- ------------- + ----------------------------------
         /4*pi\                    8/pi\            
  1 - cos|----|              64*sin |--|            
         \ 9  /                     \9 /
$$- \frac{2}{1 - \cos{\left(\frac{4 \pi}{9} \right)}} + \frac{\left(-1 + \cot^{2}{\left(\frac{\pi}{36} \right)}\right)^{2} \sin^{4}{\left(\frac{\pi}{36} \right)} \tan^{4}{\left(\frac{\pi}{9} \right)}}{64 \sin^{8}{\left(\frac{\pi}{9} \right)}}$$ 
                  2/pi\    4/2*pi\
               cos |--|*csc |----|
     2/2*pi\       \18/     \ 9  /
- csc |----| + -------------------
      \ 9  /            4
$$- \csc^{2}{\left(\frac{2 \pi}{9} \right)} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)} \csc^{4}{\left(\frac{2 \pi}{9} \right)}}{4}$$ 
                      2/2*pi\
                   cos |----|
        2              \ 9  /
- ------------- + -----------
         /4*pi\      2/-5*pi\
  1 - cos|----|   cos |-----|
         \ 9  /       \  18 /
$$- \frac{2}{1 - \cos{\left(\frac{4 \pi}{9} \right)}} + \frac{\cos^{2}{\left(\frac{2 \pi}{9} \right)}}{\cos^{2}{\left(- \frac{5 \pi}{18} \right)}}$$ 
                      2/pi\        2/pi\  
                   tan |--|     cos |--|  
  1       1            \9 /         \18/  
- - - ---------- - -------- + ------------
  2        2/pi\      4            4/2*pi\
      4*tan |--|              4*sin |----|
            \9 /                    \ 9  /
$$- \frac{1}{4 \tan^{2}{\left(\frac{\pi}{9} \right)}} - \frac{1}{2} - \frac{\tan^{2}{\left(\frac{\pi}{9} \right)}}{4} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
                  2/2*pi\
               csc |----|
     2/2*pi\       \ 9  /
- csc |----| + ----------
      \ 9  /      2/5*pi\
               csc |----|
                   \ 18 /
$$- \csc^{2}{\left(\frac{2 \pi}{9} \right)} + \frac{\csc^{2}{\left(\frac{2 \pi}{9} \right)}}{\csc^{2}{\left(\frac{5 \pi}{18} \right)}}$$ 
                  2/-5*pi\
               sec |-----|
     2/5*pi\       \  18 /
- sec |----| + -----------
      \ 18 /       2/2*pi\
                sec |----|
                    \ 9  /
$$- \sec^{2}{\left(\frac{5 \pi}{18} \right)} + \frac{\sec^{2}{\left(- \frac{5 \pi}{18} \right)}}{\sec^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                   2/4*pi\ 
                sin |----| 
      1             \ 9  / 
- ---------- + ------------
     2/7*pi\        4/2*pi\
  sin |----|   4*sin |----|
      \ 9  /         \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{7 \pi}{9} \right)}} + \frac{\sin^{2}{\left(\frac{4 \pi}{9} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
                  2/2*pi\
               cos |----|
      1            \ 9  /
- ---------- + ----------
     2/5*pi\      2/5*pi\
  cos |----|   cos |----|
      \ 18 /       \ 18 /
$$- \frac{1}{\cos^{2}{\left(\frac{5 \pi}{18} \right)}} + \frac{\cos^{2}{\left(\frac{2 \pi}{9} \right)}}{\cos^{2}{\left(\frac{5 \pi}{18} \right)}}$$ 
          4/pi\        2/2*pi\       2/4*pi\ 
       sin |--|     sin |----|    sin |----| 
  1        \9 /         \ 9  /        \ 9  / 
- - - ---------- - ----------- + ------------
  2      2/2*pi\         4/pi\        4/2*pi\
      sin |----|   16*sin |--|   4*sin |----|
          \ 9  /          \9 /         \ 9  /
$$- \frac{\sin^{2}{\left(\frac{2 \pi}{9} \right)}}{16 \sin^{4}{\left(\frac{\pi}{9} \right)}} - \frac{1}{2} - \frac{\sin^{4}{\left(\frac{\pi}{9} \right)}}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \frac{\sin^{2}{\left(\frac{4 \pi}{9} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
                    2/pi\  
                 cos |--|  
      1              \18/  
- ---------- + ------------
     2/2*pi\        4/2*pi\
  sin |----|   4*sin |----|
      \ 9  /         \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
   2/2*pi\         2      
cot |----| - -------------
    \ 9  /          /4*pi\
             1 - cos|----|
                    \ 9  /
$$- \frac{2}{1 - \cos{\left(\frac{4 \pi}{9} \right)}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
         2/2*pi\      2/7*pi\        2/pi\  
      csc |----|   csc |----|     csc |--|  
  1       \ 9  /       \ 18 /         \9 /  
- - + ---------- - ---------- - ------------
  2      2/5*pi\        2/pi\        2/7*pi\
      csc |----|   4*csc |--|   4*csc |----|
          \ 18 /         \9 /         \ 18 /
$$- \frac{\csc^{2}{\left(\frac{\pi}{9} \right)}}{4 \csc^{2}{\left(\frac{7 \pi}{18} \right)}} - \frac{1}{2} - \frac{\csc^{2}{\left(\frac{7 \pi}{18} \right)}}{4 \csc^{2}{\left(\frac{\pi}{9} \right)}} + \frac{\csc^{2}{\left(\frac{2 \pi}{9} \right)}}{\csc^{2}{\left(\frac{5 \pi}{18} \right)}}$$ 
         2/-5*pi\         2/pi\        2/-7*pi\
      sec |-----|      sec |--|     sec |-----|
  1       \  18 /          \9 /         \  18 /
- - + ----------- - ------------- - -----------
  2       2/2*pi\        2/-7*pi\         2/pi\
       sec |----|   4*sec |-----|    4*sec |--|
           \ 9  /         \  18 /          \9 /
$$- \frac{\sec^{2}{\left(- \frac{7 \pi}{18} \right)}}{4 \sec^{2}{\left(\frac{\pi}{9} \right)}} - \frac{1}{2} - \frac{\sec^{2}{\left(\frac{\pi}{9} \right)}}{4 \sec^{2}{\left(- \frac{7 \pi}{18} \right)}} + \frac{\sec^{2}{\left(- \frac{5 \pi}{18} \right)}}{\sec^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                       2/4*pi\ 
                    sin |----| 
        2               \ 9  / 
- -------------- + ------------
         /17*pi\        4/2*pi\
  1 - sin|-----|   4*sin |----|
         \  18 /         \ 9  /
$$- \frac{2}{1 - \sin{\left(\frac{17 \pi}{18} \right)}} + \frac{\sin^{2}{\left(\frac{4 \pi}{9} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
    1             2     
---------- - -----------
   2/2*pi\          /pi\
tan |----|   1 - sin|--|
    \ 9  /          \18/
$$- \frac{2}{1 - \sin{\left(\frac{\pi}{18} \right)}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                           2
             /       2/pi\\ 
             |1 + tan |--|| 
    1        \        \9 // 
---------- - ---------------
   2/2*pi\           2/pi\  
tan |----|      4*tan |--|  
    \ 9  /            \9 /
$$- \frac{\left(\tan^{2}{\left(\frac{\pi}{9} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{\pi}{9} \right)}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                             2
             /       2/7*pi\\ 
             |1 + tan |----|| 
    1        \        \ 18 // 
---------- - -----------------
   2/2*pi\           2/7*pi\  
tan |----|      4*tan |----|  
    \ 9  /            \ 18 /
$$- \frac{\left(1 + \tan^{2}{\left(\frac{7 \pi}{18} \right)}\right)^{2}}{4 \tan^{2}{\left(\frac{7 \pi}{18} \right)}} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                     2/pi\  
                  cos |--|  
       2              \18/  
- ----------- + ------------
         /pi\        4/2*pi\
  1 - sin|--|   4*sin |----|
         \18/         \ 9  /
$$- \frac{2}{1 - \sin{\left(\frac{\pi}{18} \right)}} + \frac{\cos^{2}{\left(\frac{\pi}{18} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
   2/2*pi\            2         
cot |----| - -------------------
    \ 9  /               2/2*pi\
                 -1 + cot |----|
                          \ 9  /
             1 - ---------------
                         2/2*pi\
                  1 + cot |----|
                          \ 9  /
$$- \frac{2}{- \frac{-1 + \cot^{2}{\left(\frac{2 \pi}{9} \right)}}{1 + \cot^{2}{\left(\frac{2 \pi}{9} \right)}} + 1} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
                                   2/pi\
                                tan |--|
  1      2/2*pi\       1            \9 /
- - + cot |----| - ---------- - --------
  2       \ 9  /        2/pi\      4    
                   4*tan |--|           
                         \9 /
$$- \frac{1}{4 \tan^{2}{\left(\frac{\pi}{9} \right)}} - \frac{1}{2} - \frac{\tan^{2}{\left(\frac{\pi}{9} \right)}}{4} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
                   2/5*pi\ 
                sin |----| 
      1             \ 9  / 
- ---------- + ------------
     2/2*pi\        4/2*pi\
  sin |----|   4*sin |----|
      \ 9  /         \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \frac{\sin^{2}{\left(\frac{5 \pi}{9} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
                2                 4                2
  /       2/pi\\    /       2/pi\\  /        2/pi\\ 
  |1 + cot |--||    |1 + cot |--|| *|-1 + cot |--|| 
  \        \9 //    \        \9 //  \         \36// 
- --------------- + --------------------------------
          2/pi\                        2            
     4*cot |--|          /       2/pi\\     4/pi\   
           \9 /       64*|1 + cot |--|| *cot |--|   
                         \        \36//      \9 /
$$- \frac{\left(1 + \cot^{2}{\left(\frac{\pi}{9} \right)}\right)^{2}}{4 \cot^{2}{\left(\frac{\pi}{9} \right)}} + \frac{\left(-1 + \cot^{2}{\left(\frac{\pi}{36} \right)}\right)^{2} \left(1 + \cot^{2}{\left(\frac{\pi}{9} \right)}\right)^{4}}{64 \left(1 + \cot^{2}{\left(\frac{\pi}{36} \right)}\right)^{2} \cot^{4}{\left(\frac{\pi}{9} \right)}}$$ 
    1           2/5*pi\
---------- - sec |----|
   2/2*pi\       \ 18 /
tan |----|             
    \ 9  /
$$- \sec^{2}{\left(\frac{5 \pi}{18} \right)} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                                            4               2
                      2/pi\   /       2/pi\\  /       2/pi\\ 
                   tan |--|   |1 + tan |--|| *|1 - tan |--|| 
  1       1            \9 /   \        \9 //  \        \36// 
- - - ---------- - -------- + -------------------------------
  2        2/pi\      4                          2           
      4*tan |--|                   /       2/pi\\     4/pi\  
            \9 /                64*|1 + tan |--|| *tan |--|  
                                   \        \36//      \9 /
$$- \frac{1}{4 \tan^{2}{\left(\frac{\pi}{9} \right)}} - \frac{1}{2} - \frac{\tan^{2}{\left(\frac{\pi}{9} \right)}}{4} + \frac{\left(1 - \tan^{2}{\left(\frac{\pi}{36} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{\pi}{9} \right)} + 1\right)^{4}}{64 \left(\tan^{2}{\left(\frac{\pi}{36} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{\pi}{9} \right)}}$$ 
                   4/-5*pi\
                sec |-----|
     2/-5*pi\       \  18 /
- sec |-----| + -----------
      \  18 /         2/pi\
                 4*sec |--|
                       \18/
$$- \sec^{2}{\left(- \frac{5 \pi}{18} \right)} + \frac{\sec^{4}{\left(- \frac{5 \pi}{18} \right)}}{4 \sec^{2}{\left(\frac{\pi}{18} \right)}}$$ 
                   2/4*pi\ 
                sin |----| 
      1             \ 9  / 
- ---------- + ------------
     2/2*pi\        4/2*pi\
  sin |----|   4*sin |----|
      \ 9  /         \ 9  /
$$- \frac{1}{\sin^{2}{\left(\frac{2 \pi}{9} \right)}} + \frac{\sin^{2}{\left(\frac{4 \pi}{9} \right)}}{4 \sin^{4}{\left(\frac{2 \pi}{9} \right)}}$$ 
                             2
             /       2/7*pi\\ 
             |1 + cot |----|| 
   2/2*pi\   \        \ 18 // 
cot |----| - -----------------
    \ 9  /           2/7*pi\  
                4*cot |----|  
                      \ 18 /
$$- \frac{\left(\cot^{2}{\left(\frac{7 \pi}{18} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{7 \pi}{18} \right)}} + \cot^{2}{\left(\frac{2 \pi}{9} \right)}$$ 
                                   2/pi\
                                tan |--|
  1       1            1            \9 /
- - + ---------- - ---------- - --------
  2      2/2*pi\        2/pi\      4    
      tan |----|   4*tan |--|           
          \ 9  /         \9 /
$$- \frac{1}{4 \tan^{2}{\left(\frac{\pi}{9} \right)}} - \frac{1}{2} - \frac{\tan^{2}{\left(\frac{\pi}{9} \right)}}{4} + \frac{1}{\tan^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
                  2/5*pi\
               sec |----|
     2/5*pi\       \ 18 /
- sec |----| + ----------
      \ 18 /      2/2*pi\
               sec |----|
                   \ 9  /
$$- \sec^{2}{\left(\frac{5 \pi}{18} \right)} + \frac{\sec^{2}{\left(\frac{5 \pi}{18} \right)}}{\sec^{2}{\left(\frac{2 \pi}{9} \right)}}$$ 
-sec(5*pi/18)^2 + sec(5*pi/18)^2/sec(2*pi/9)^2
    © Sr Examen