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