Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(8*x)/((2*sin(4*x)*cos(4*x)))
¿Cómo vas a descomponer esta cos(8*x)/((2*sin(4*x)*cos(4*x))) expresión en fracciones?
Solución
Ha introducido
[src]
cos(8*x) ------------------- 2*sin(4*x)*cos(4*x)
$$\frac{\cos{\left(8 x \right)}}{2 \sin{\left(4 x \right)} \cos{\left(4 x \right)}}$$
cos(8*x)/(((2*sin(4*x))*cos(4*x)))
Potencias
[src]
/ -8*I*x 8*I*x\
|e e |
I*|------- + ------|
\ 2 2 /
---------------------------------------
/ -4*I*x 4*I*x\
|e e | / -4*I*x 4*I*x\
|------- + ------|*\- e + e /
\ 2 2 /
$$\frac{i \left(\frac{e^{8 i x}}{2} + \frac{e^{- 8 i x}}{2}\right)}{\left(\frac{e^{4 i x}}{2} + \frac{e^{- 4 i x}}{2}\right) \left(e^{4 i x} - e^{- 4 i x}\right)}$$
i*(exp(-8*i*x)/2 + exp(8*i*x)/2)/((exp(-4*i*x)/2 + exp(4*i*x)/2)*(-exp(-4*i*x) + exp(4*i*x)))
Parte trigonométrica
[src]
/ 2 \
2/ pi\ | cos (4*x) |
cos |4*x - --|*|-1 + --------------|
\ 2 / | 2/ pi\|
| cos |4*x - --||
\ \ 2 //
-----------------------------------------------
/ 2 \
| cos (2*x) | 3/ pi\
4*|-1 + --------------|*cos(2*x)*cos |2*x - --|
| 2/ pi\| \ 2 /
| cos |2*x - --||
\ \ 2 //
$$\frac{\left(\frac{\cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(4 x - \frac{\pi}{2} \right)}}{4 \left(\frac{\cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(2 x \right)} \cos^{3}{\left(2 x - \frac{\pi}{2} \right)}}$$
/ 2 \
\1 - tan (4*x)/*csc(2*x)
------------------------------------------
/ 2 \ 3 2/pi \
4*\1 - tan (2*x)/*cos (2*x)*csc |-- - 4*x|
\2 /
$$\frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \csc{\left(2 x \right)}}{4 \left(1 - \tan^{2}{\left(2 x \right)}\right) \cos^{3}{\left(2 x \right)} \csc^{2}{\left(- 4 x + \frac{\pi}{2} \right)}}$$
2
/ 2 \ 4 / 2 \
\-1 + cot (2*x)/ *sin (2*x)*\1 - tan (4*x)/*cot(2*x)
----------------------------------------------------
4
/ 2 \ / 2 \ 8
4*\1 - tan (2*x)/*\-1 + cot (x)/ *sin (x)
$$\frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \left(\cot^{2}{\left(2 x \right)} - 1\right)^{2} \sin^{4}{\left(2 x \right)} \cot{\left(2 x \right)}}{4 \left(1 - \tan^{2}{\left(2 x \right)}\right) \left(\cot^{2}{\left(x \right)} - 1\right)^{4} \sin^{8}{\left(x \right)}}$$
/ 2 \
3 | sec (4*x) | / pi\
sec (2*x)*|1 - --------------|*sec|2*x - --|
| 2/ pi\| \ 2 /
| sec |4*x - --||
\ \ 2 //
--------------------------------------------
/ 2 \
| sec (2*x) | 2
4*|1 - --------------|*sec (4*x)
| 2/ pi\|
| sec |2*x - --||
\ \ 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(4 x \right)}}{\sec^{2}{\left(4 x - \frac{\pi}{2} \right)}} + 1\right) \sec^{3}{\left(2 x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}{4 \left(- \frac{\sec^{2}{\left(2 x \right)}}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(4 x \right)}}$$
3 2 / 1 \
csc (2*x)*sin (4*x)*|-1 + ---------|
| 2 |
\ tan (4*x)/
------------------------------------
/ 1 \
4*|-1 + ---------|*cos(2*x)
| 2 |
\ tan (2*x)/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \sin^{2}{\left(4 x \right)} \csc^{3}{\left(2 x \right)}}{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)}}$$
/ 4 \
2/pi \ | 4*sin (4*x)|
sin |-- + 4*x|*|1 - -----------|*sin(4*x)
\2 / | 2 |
\ sin (8*x) /
--------------------------------------------
/ 4 \
| 4*sin (2*x)| 2 4/pi \
8*|1 - -----------|*sin (2*x)*sin |-- + 2*x|
| 2 | \2 /
\ sin (4*x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(4 x \right)}}{\sin^{2}{\left(8 x \right)}} + 1\right) \sin{\left(4 x \right)} \sin^{2}{\left(4 x + \frac{\pi}{2} \right)}}{8 \left(- \frac{4 \sin^{4}{\left(2 x \right)}}{\sin^{2}{\left(4 x \right)}} + 1\right) \sin^{2}{\left(2 x \right)} \sin^{4}{\left(2 x + \frac{\pi}{2} \right)}}$$
/pi \
csc(4*x)*csc|-- - 4*x|
\2 /
----------------------
/pi \
2*csc|-- - 8*x|
\2 /
$$\frac{\csc{\left(4 x \right)} \csc{\left(- 4 x + \frac{\pi}{2} \right)}}{2 \csc{\left(- 8 x + \frac{\pi}{2} \right)}}$$
/pi \
sec(4*x)*sec|-- - 4*x|
\2 /
----------------------
2*sec(8*x)
$$\frac{\sec{\left(4 x \right)} \sec{\left(- 4 x + \frac{\pi}{2} \right)}}{2 \sec{\left(8 x \right)}}$$
/ 2 \ / 2/ pi\\ / pi\
\1 + cot (2*x)/*|1 + cot |2*x + --||*cot|4*x + --|
\ \ 4 // \ 4 /
--------------------------------------------------
/ 2/ pi\\ / pi\
4*|1 + cot |4*x + --||*cot(2*x)*cot|2*x + --|
\ \ 4 // \ 4 /
$$\frac{\left(\cot^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \cot{\left(4 x + \frac{\pi}{4} \right)}}{4 \left(\cot^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \cot{\left(2 x \right)} \cot{\left(2 x + \frac{\pi}{4} \right)}}$$
2
/ 2 \ / 2 \
\1 + cot (2*x)/ *\-1 + cot (4*x)/
-------------------------------------------
/ 2 \ / 2 \
4*\1 + cot (4*x)/*\-1 + cot (2*x)/*cot(2*x)
$$\frac{\left(\cot^{2}{\left(2 x \right)} + 1\right)^{2} \left(\cot^{2}{\left(4 x \right)} - 1\right)}{4 \left(\cot^{2}{\left(2 x \right)} - 1\right) \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot{\left(2 x \right)}}$$
2 / 1 \
sin (4*x)*|-1 + ---------|*sec(2*x)
| 2 |
\ tan (4*x)/
-----------------------------------
/ 1 \ 3
4*|-1 + ---------|*sin (2*x)
| 2 |
\ tan (2*x)/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \sin^{2}{\left(4 x \right)} \sec{\left(2 x \right)}}{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \sin^{3}{\left(2 x \right)}}$$
/ 2/ pi\\
| cos |4*x - --||
2 | \ 2 /|
cos (4*x)*|1 - --------------|
| 2 |
\ cos (4*x) /
----------------------------------------------
/ 2/ pi\\
| cos |2*x - --||
| \ 2 /| 3 / pi\
4*|1 - --------------|*cos (2*x)*cos|2*x - --|
| 2 | \ 2 /
\ cos (2*x) /
$$\frac{\left(1 - \frac{\cos^{2}{\left(4 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(4 x \right)}}\right) \cos^{2}{\left(4 x \right)}}{4 \left(1 - \frac{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 x \right)}}\right) \cos^{3}{\left(2 x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)}}$$
cos(8*x)
------------------------
/ pi\
2*cos(4*x)*cos|4*x - --|
\ 2 /
$$\frac{\cos{\left(8 x \right)}}{2 \cos{\left(4 x \right)} \cos{\left(4 x - \frac{\pi}{2} \right)}}$$
/ 2 \ / 2/ pi\\ / pi\
\1 + tan (2*x)/*|1 + tan |2*x + --||*tan|4*x + --|
\ \ 4 // \ 4 /
--------------------------------------------------
/ 2/ pi\\ / pi\
4*|1 + tan |4*x + --||*tan(2*x)*tan|2*x + --|
\ \ 4 // \ 4 /
$$\frac{\left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \tan{\left(4 x + \frac{\pi}{4} \right)}}{4 \left(\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \tan{\left(2 x \right)} \tan{\left(2 x + \frac{\pi}{4} \right)}}$$
2/ pi\ / pi\ / pi\
sin |4*x + --|*cot|4*x + --|*tan(2*x)*tan|2*x + --|
\ 4 / \ 4 / \ 4 /
---------------------------------------------------
2 2/ pi\
4*sin (2*x)*sin |2*x + --|
\ 4 /
$$\frac{\sin^{2}{\left(4 x + \frac{\pi}{4} \right)} \tan{\left(2 x \right)} \tan{\left(2 x + \frac{\pi}{4} \right)} \cot{\left(4 x + \frac{\pi}{4} \right)}}{4 \sin^{2}{\left(2 x \right)} \sin^{2}{\left(2 x + \frac{\pi}{4} \right)}}$$
/ 4 \
2/pi \ | 4*sin (4*x)|
sin |-- + 4*x|*|1 - -----------|*sin(4*x)
\2 / | 2 |
\ sin (8*x) /
-----------------------------------------
/ 4 \
| 4*sin (2*x)| 4 2
8*|1 - -----------|*cos (2*x)*sin (2*x)
| 2 |
\ sin (4*x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(4 x \right)}}{\sin^{2}{\left(8 x \right)}} + 1\right) \sin{\left(4 x \right)} \sin^{2}{\left(4 x + \frac{\pi}{2} \right)}}{8 \left(- \frac{4 \sin^{4}{\left(2 x \right)}}{\sin^{2}{\left(4 x \right)}} + 1\right) \sin^{2}{\left(2 x \right)} \cos^{4}{\left(2 x \right)}}$$
4 / 2 \ / 2 \
cos (2*x)*\1 - tan (2*x)/*\1 - tan (4*x)/
-----------------------------------------
4
/ 2 \ 8
4*\1 - tan (x)/ *cos (x)*tan(2*x)
$$\frac{\left(1 - \tan^{2}{\left(2 x \right)}\right) \left(1 - \tan^{2}{\left(4 x \right)}\right) \cos^{4}{\left(2 x \right)}}{4 \left(1 - \tan^{2}{\left(x \right)}\right)^{4} \cos^{8}{\left(x \right)} \tan{\left(2 x \right)}}$$
/ 1 \
|1 - ---------|*tan(2*x)
| 2 |
\ tan (4*x)/
------------------------
2
-1 + tan (2*x)
$$\frac{\left(1 - \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} - 1}$$
2 / 2 \
cos (4*x)*\1 - tan (4*x)/
------------------------------------
/ 2 \ 3
4*\1 - tan (2*x)/*cos (2*x)*sin(2*x)
$$\frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \cos^{2}{\left(4 x \right)}}{4 \left(1 - \tan^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)} \cos^{3}{\left(2 x \right)}}$$
/ 2/pi \\
| csc |-- - 4*x||
3/pi \ | \2 /|
csc |-- - 2*x|*|1 - --------------|*csc(2*x)
\2 / | 2 |
\ csc (4*x) /
--------------------------------------------
/ 2/pi \\
| csc |-- - 2*x||
| \2 /| 2/pi \
4*|1 - --------------|*csc |-- - 4*x|
| 2 | \2 /
\ csc (2*x) /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- 4 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(4 x \right)}}\right) \csc{\left(2 x \right)} \csc^{3}{\left(- 2 x + \frac{\pi}{2} \right)}}{4 \left(1 - \frac{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 x \right)}}\right) \csc^{2}{\left(- 4 x + \frac{\pi}{2} \right)}}$$
4
/ 2 \ / 2 \
\1 + cot (x)/ *\-1 + cot (4*x)/*cot(2*x)
--------------------------------------------
2
/ 2 \ / 2 \ 4
16*\1 + cot (2*x)/ *\-1 + cot (2*x)/*cot (x)
$$\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{4} \left(\cot^{2}{\left(4 x \right)} - 1\right) \cot{\left(2 x \right)}}{16 \left(\cot^{2}{\left(2 x \right)} - 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)^{2} \cot^{4}{\left(x \right)}}$$
4
/ 2 \ 3 / 1 \
\1 + tan (x)/ *tan (2*x)*|-1 + ---------|
| 2 |
\ tan (4*x)/
--------------------------------------------
2
/ 2 \ / 1 \ 4
16*\1 + tan (2*x)/ *|-1 + ---------|*tan (x)
| 2 |
\ tan (2*x)/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)^{4} \tan^{3}{\left(2 x \right)}}{16 \left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \tan^{4}{\left(x \right)}}$$
4 2
/ 2 \ / 2 \ / 1 \
\1 + cot (x)/ *\-1 + cot (2*x)/ *|1 - ---------|*cot(2*x)
| 2 |
\ cot (4*x)/
---------------------------------------------------------
2 4
/ 2 \ / 1 \ / 2 \
4*\1 + cot (2*x)/ *|1 - ---------|*\-1 + cot (x)/
| 2 |
\ cot (2*x)/
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(4 x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{4} \left(\cot^{2}{\left(2 x \right)} - 1\right)^{2} \cot{\left(2 x \right)}}{4 \left(1 - \frac{1}{\cot^{2}{\left(2 x \right)}}\right) \left(\cot^{2}{\left(x \right)} - 1\right)^{4} \left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}$$
/ 2 \ / 2/ pi\\ / pi\
\1 + cot (2*x)/*|1 + tan |2*x + --||*tan|4*x + --|
\ \ 4 // \ 4 /
--------------------------------------------------
/ 2/ pi\\ / pi\
4*|1 + tan |4*x + --||*cot(2*x)*tan|2*x + --|
\ \ 4 // \ 4 /
$$\frac{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) \tan{\left(4 x + \frac{\pi}{4} \right)}}{4 \left(\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \tan{\left(2 x + \frac{\pi}{4} \right)} \cot{\left(2 x \right)}}$$
2 / 2 \
sin (4*x)*\-1 + cot (4*x)/*tan(2*x)
-----------------------------------
/ 2 \ 4
4*\-1 + cot (2*x)/*sin (2*x)
$$\frac{\left(\cot^{2}{\left(4 x \right)} - 1\right) \sin^{2}{\left(4 x \right)} \tan{\left(2 x \right)}}{4 \left(\cot^{2}{\left(2 x \right)} - 1\right) \sin^{4}{\left(2 x \right)}}$$
/ 2/ pi\\
| sec |4*x - --||
3/ pi\ | \ 2 /|
sec |2*x - --|*|-1 + --------------|*sec(2*x)
\ 2 / | 2 |
\ sec (4*x) /
---------------------------------------------
/ 2/ pi\\
| sec |2*x - --||
| \ 2 /| 2/ pi\
4*|-1 + --------------|*sec |4*x - --|
| 2 | \ 2 /
\ sec (2*x) /
$$\frac{\left(-1 + \frac{\sec^{2}{\left(4 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(4 x \right)}}\right) \sec{\left(2 x \right)} \sec^{3}{\left(2 x - \frac{\pi}{2} \right)}}{4 \left(-1 + \frac{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 x \right)}}\right) \sec^{2}{\left(4 x - \frac{\pi}{2} \right)}}$$
2
/ 2 \ / 2 \
\1 + tan (2*x)/ *\1 - tan (4*x)/
------------------------------------------
/ 2 \ / 2 \
4*\1 + tan (4*x)/*\1 - tan (2*x)/*tan(2*x)
$$\frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}}{4 \left(1 - \tan^{2}{\left(2 x \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(2 x \right)}}$$
/ 2/ pi\\ / pi\ |1 + tan |2*x + --||*tan(2*x)*tan|4*x + --| \ \ 4 // \ 4 / ---------------------------------------------- / 2/ pi\\ 2 / pi\ 4*|1 + tan |4*x + --||*sin (2*x)*tan|2*x + --| \ \ 4 // \ 4 /
$$\frac{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \tan{\left(2 x \right)} \tan{\left(4 x + \frac{\pi}{4} \right)}}{4 \left(\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(2 x \right)} \tan{\left(2 x + \frac{\pi}{4} \right)}}$$
4
/ 2 \ / 2 \ / 2 \
\1 + tan (x)/ *\1 - tan (2*x)/*\1 - tan (4*x)/
----------------------------------------------
2 4
/ 2 \ / 2 \
4*\1 + tan (2*x)/ *\1 - tan (x)/ *tan(2*x)
$$\frac{\left(1 - \tan^{2}{\left(2 x \right)}\right) \left(1 - \tan^{2}{\left(4 x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)^{4}}{4 \left(1 - \tan^{2}{\left(x \right)}\right)^{4} \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \tan{\left(2 x \right)}}$$
3 / 2 \
sec (2*x)*\1 - tan (4*x)/
------------------------------------
/ 2 \ 2
4*\1 - tan (2*x)/*sec (4*x)*sin(2*x)
$$\frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \sec^{3}{\left(2 x \right)}}{4 \left(1 - \tan^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)} \sec^{2}{\left(4 x \right)}}$$
/ pi\
sec(4*x)*sec|4*x - --|
\ 2 /
----------------------
2*sec(8*x)
$$\frac{\sec{\left(4 x \right)} \sec{\left(4 x - \frac{\pi}{2} \right)}}{2 \sec{\left(8 x \right)}}$$
2 / 2 \
cos (4*x)*\1 - tan (4*x)/
------------------------------------
/ 2 \ 4
4*\1 - tan (2*x)/*cos (2*x)*tan(2*x)
$$\frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \cos^{2}{\left(4 x \right)}}{4 \left(1 - \tan^{2}{\left(2 x \right)}\right) \cos^{4}{\left(2 x \right)} \tan{\left(2 x \right)}}$$
sec(4*x) ------------------- 2*sec(8*x)*sin(4*x)
$$\frac{\sec{\left(4 x \right)}}{2 \sin{\left(4 x \right)} \sec{\left(8 x \right)}}$$
/pi \
sin|-- + 8*x|
\2 /
------------------------
/pi \
2*sin(4*x)*sin|-- + 4*x|
\2 /
$$\frac{\sin{\left(8 x + \frac{\pi}{2} \right)}}{2 \sin{\left(4 x \right)} \sin{\left(4 x + \frac{\pi}{2} \right)}}$$
/ 2/ pi\\ / pi\
|1 + tan |2*x + --||*tan|4*x + --|
\ \ 4 // \ 4 /
-------------------------------------------------------
/ 2/ pi\\ 2 / pi\
4*|1 + tan |4*x + --||*cos (2*x)*tan(2*x)*tan|2*x + --|
\ \ 4 // \ 4 /
$$\frac{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \tan{\left(4 x + \frac{\pi}{4} \right)}}{4 \left(\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \cos^{2}{\left(2 x \right)} \tan{\left(2 x \right)} \tan{\left(2 x + \frac{\pi}{4} \right)}}$$
/ 2 \ | sin (8*x) | |-1 + -----------|*sin(4*x) | 4 | \ 4*sin (4*x)/ ------------------------------ / 2 \ | sin (4*x) | 2 2*|-1 + -----------|*sin (2*x) | 4 | \ 4*sin (2*x)/
$$\frac{\left(-1 + \frac{\sin^{2}{\left(8 x \right)}}{4 \sin^{4}{\left(4 x \right)}}\right) \sin{\left(4 x \right)}}{2 \left(-1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}\right) \sin^{2}{\left(2 x \right)}}$$
/ 2 \
3 | csc (4*x) | /pi \
csc (2*x)*|-1 + --------------|*csc|-- - 2*x|
| 2/pi \| \2 /
| csc |-- - 4*x||
\ \2 //
---------------------------------------------
/ 2 \
| csc (2*x) | 2
4*|-1 + --------------|*csc (4*x)
| 2/pi \|
| csc |-- - 2*x||
\ \2 //
$$\frac{\left(\frac{\csc^{2}{\left(4 x \right)}}{\csc^{2}{\left(- 4 x + \frac{\pi}{2} \right)}} - 1\right) \csc^{3}{\left(2 x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}{4 \left(\frac{\csc^{2}{\left(2 x \right)}}{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} - 1\right) \csc^{2}{\left(4 x \right)}}$$
csc(4*x)*sec(4*x)
-----------------
2*sec(8*x)
$$\frac{\csc{\left(4 x \right)} \sec{\left(4 x \right)}}{2 \sec{\left(8 x \right)}}$$
2 / 1 \
sin (4*x)*|-1 + ---------|
| 2 |
\ tan (4*x)/
-------------------------------------
/ 1 \ 3
4*|-1 + ---------|*cos(2*x)*sin (2*x)
| 2 |
\ tan (2*x)/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \sin^{2}{\left(4 x \right)}}{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \sin^{3}{\left(2 x \right)} \cos{\left(2 x \right)}}$$
sin(4*x)^2*(-1 + tan(4*x)^(-2))/(4*(-1 + tan(2*x)^(-2))*cos(2*x)*sin(2*x)^3)