Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(pi/(2-3*a))^2-(-cos(3*a))^2
¿Cómo vas a descomponer esta cos(pi/(2-3*a))^2-(-cos(3*a))^2 expresión en fracciones?
Solución
Ha introducido
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2/ pi \ 2
cos |-------| - (-cos(3*a))
\2 - 3*a/
$$- \left(- \cos{\left(3 a \right)}\right)^{2} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
cos(pi/(2 - 3*a))^2 - (-cos(3*a))^2
Simplificación general
[src]
2/ pi \ 2
cos |--------| - cos (3*a)
\-2 + 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{3 a - 2} \right)}$$
cos(pi/(-2 + 3*a))^2 - cos(3*a)^2
Combinatoria
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/ / pi \ \ / / pi \\ -|- cos|-------| + cos(3*a)|*|cos(3*a) + cos|-------|| \ \2 - 3*a/ / \ \2 - 3*a//
$$- \left(\cos{\left(3 a \right)} - \cos{\left(\frac{\pi}{2 - 3 a} \right)}\right) \left(\cos{\left(3 a \right)} + \cos{\left(\frac{\pi}{2 - 3 a} \right)}\right)$$
-(-cos(pi/(2 - 3*a)) + cos(3*a))*(cos(3*a) + cos(pi/(2 - 3*a)))
Unión de expresiones racionales
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2/ pi \ 2
cos |-------| - cos (3*a)
\2 - 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
cos(pi/(2 - 3*a))^2 - cos(3*a)^2
Denominador común
[src]
2/ pi \ 2
cos |-------| - cos (3*a)
\2 - 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
cos(pi/(2 - 3*a))^2 - cos(3*a)^2
Denominador racional
[src]
2/ pi \ 2
cos |-------| - cos (3*a)
\2 - 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
cos(pi/(2 - 3*a))^2 - cos(3*a)^2
Potencias
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2 / pi*I -pi*I \ | ------- -------| 2 | 2 - 3*a 2 - 3*a| / -3*I*a 3*I*a\ |e e | | e e | |-------- + --------| - |- ------- - ------| \ 2 2 / \ 2 2 /
$$- \left(- \frac{e^{3 i a}}{2} - \frac{e^{- 3 i a}}{2}\right)^{2} + \left(\frac{e^{\frac{i \pi}{2 - 3 a}}}{2} + \frac{e^{- \frac{i \pi}{2 - 3 a}}}{2}\right)^{2}$$
2/ pi \ 2
cos |-------| - cos (3*a)
\2 - 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
cos(pi/(2 - 3*a))^2 - cos(3*a)^2
Compilar la expresión
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2/ pi \ 2
cos |-------| - cos (3*a)
\2 - 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
cos(pi/(2 - 3*a))^2 - cos(3*a)^2
Parte trigonométrica
[src]
1 1
-------------- - ---------------------
2/ pi\ 2/ pi pi \
sec |3*a - --| sec |- -- + --------|
\ 2 / \ 2 -2 + 3*a/
$$\frac{1}{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{3 a - 2} \right)}}$$
2
/ 2/ pi \ \
| csc |-----------| |
| \2*(2 - 3*a)/ |
|-1 + ----------------------|
| 2/pi pi \|
| csc |-- - -----------||
1 1 \ \2 2*(2 - 3*a)//
- - - --------------- + ------------------------------
2 /pi \ 4/ pi \
2*csc|-- - 6*a| csc |--------|
\2 / \-4 + 6*a/
$$\frac{\left(\frac{\csc^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)}}{\csc^{2}{\left(\frac{\pi}{2} - \frac{\pi}{2 \left(2 - 3 a\right)} \right)}} - 1\right)^{2}}{\csc^{4}{\left(\frac{\pi}{6 a - 4} \right)}} - \frac{1}{2} - \frac{1}{2 \csc{\left(- 6 a + \frac{\pi}{2} \right)}}$$
2/ pi \ 2/3*a\
4*tan |------------| 4*tan |---|
\2*(-2 + 3*a)/ \ 2 /
- ------------------------- + ----------------
2 2
/ 2/ pi \\ / 2/3*a\\
|1 + tan |------------|| |1 + tan |---||
\ \2*(-2 + 3*a)// \ \ 2 //
$$- \frac{4 \tan^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)}}{\left(\tan^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{3 a}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}}$$
2 2/ pi \
sin (3*a) - sin |--------|
\-2 + 3*a/
$$\sin^{2}{\left(3 a \right)} - \sin^{2}{\left(\frac{\pi}{3 a - 2} \right)}$$
1 1
------------------- - --------------
2/pi pi \ 2/pi \
csc |-- - --------| csc |-- - 3*a|
\2 -2 + 3*a/ \2 /
$$- \frac{1}{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(\frac{\pi}{2} - \frac{\pi}{3 a - 2} \right)}}$$
2
/ 2/ pi \ \
| cos |-----------| |
1 cos(6*a) | \2*(2 - 3*a)/ | 4/ pi pi \
- - - -------- + |-1 + ------------------------| *cos |- -- + --------|
2 2 | 2/ pi pi \| \ 2 -4 + 6*a/
| cos |- -- + -----------||
\ \ 2 2*(2 - 3*a)//
$$\left(\frac{\cos^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(2 - 3 a\right)} \right)}} - 1\right)^{2} \cos^{4}{\left(- \frac{\pi}{2} + \frac{\pi}{6 a - 4} \right)} - \frac{\cos{\left(6 a \right)}}{2} - \frac{1}{2}$$
1 1
------------- - ---------
2/ pi \ 2
sec |-------| sec (3*a)
\2 - 3*a/
$$\frac{1}{\sec^{2}{\left(\frac{\pi}{2 - 3 a} \right)}} - \frac{1}{\sec^{2}{\left(3 a \right)}}$$
2/pi pi \ 2/pi \
sin |-- + -------| - sin |-- + 3*a|
\2 2 - 3*a/ \2 /
$$\sin^{2}{\left(\frac{\pi}{2} + \frac{\pi}{2 - 3 a} \right)} - \sin^{2}{\left(3 a + \frac{\pi}{2} \right)}$$
2
/ 2/ pi \ \
| sin |-------| |
1 cos(6*a) | \2 - 3*a/ | 4/ pi \
- - - -------- + |-1 + ----------------| *sin |--------|
2 2 | 4/ pi \| \-4 + 6*a/
| 4*sin |--------||
\ \-4 + 6*a//
$$\left(\frac{\sin^{2}{\left(\frac{\pi}{2 - 3 a} \right)}}{4 \sin^{4}{\left(\frac{\pi}{6 a - 4} \right)}} - 1\right)^{2} \sin^{4}{\left(\frac{\pi}{6 a - 4} \right)} - \frac{\cos{\left(6 a \right)}}{2} - \frac{1}{2}$$
2 2/ pi \ 4/ pi \
sin (3*a) - 4*sin |--------| + 4*sin |--------|
\-4 + 6*a/ \-4 + 6*a/
$$\sin^{2}{\left(3 a \right)} + 4 \sin^{4}{\left(\frac{\pi}{6 a - 4} \right)} - 4 \sin^{2}{\left(\frac{\pi}{6 a - 4} \right)}$$
2/ pi \ 2/3*a\
4*cot |------------| 4*cot |---|
\2*(-2 + 3*a)/ \ 2 /
- ------------------------- + ----------------
2 2
/ 2/ pi \\ / 2/3*a\\
|1 + cot |------------|| |1 + cot |---||
\ \2*(-2 + 3*a)// \ \ 2 //
$$- \frac{4 \cot^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)}}{\left(\cot^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{3 a}{2} \right)}}{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}}$$
2/ pi \ 2
cos |--------| - cos (3*a)
\-2 + 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{3 a - 2} \right)}$$
2/ pi\ 2/ pi pi \
cos |3*a - --| - cos |- -- + --------|
\ 2 / \ 2 -2 + 3*a/
$$- \cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{3 a - 2} \right)} + \cos^{2}{\left(3 a - \frac{\pi}{2} \right)}$$
1 1
--------- - --------------
2 2/ pi \
csc (3*a) csc |--------|
\-2 + 3*a/
$$- \frac{1}{\csc^{2}{\left(\frac{\pi}{3 a - 2} \right)}} + \frac{1}{\csc^{2}{\left(3 a \right)}}$$
2 2
/ 2/ pi \\ / 2/3*a\\
|1 - tan |------------|| |1 - tan |---||
\ \2*(-2 + 3*a)// \ \ 2 //
------------------------- - ----------------
2 2
/ 2/ pi \\ / 2/3*a\\
|1 + tan |------------|| |1 + tan |---||
\ \2*(-2 + 3*a)// \ \ 2 //
$$- \frac{\left(1 - \tan^{2}{\left(\frac{3 a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}} + \frac{\left(1 - \tan^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)} + 1\right)^{2}}$$
2
1 + cos(6*a) / 2/ pi \\ 4/ pi \
- ------------ + |-1 + cot |------------|| *sin |--------|
2 \ \2*(-2 + 3*a)// \-4 + 6*a/
$$- \frac{\cos{\left(6 a \right)} + 1}{2} + \left(\cot^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{\pi}{6 a - 4} \right)}$$
2
/ 2/ pi \\ 4/ pi \
2 16*|-1 + cot |-----------|| *cot |------------|
1 -1 + cot (3*a) \ \2*(2 - 3*a)// \2*(-4 + 6*a)/
- - - ----------------- + -----------------------------------------------
2 / 2 \ 4
2*\1 + cot (3*a)/ / 2/ pi \\
|1 + cot |------------||
\ \2*(-4 + 6*a)//
$$- \frac{\cot^{2}{\left(3 a \right)} - 1}{2 \left(\cot^{2}{\left(3 a \right)} + 1\right)} + \frac{16 \left(\cot^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)} - 1\right)^{2} \cot^{4}{\left(\frac{\pi}{2 \left(6 a - 4\right)} \right)}}{\left(\cot^{2}{\left(\frac{\pi}{2 \left(6 a - 4\right)} \right)} + 1\right)^{4}} - \frac{1}{2}$$
2 2
/ 2/ pi \\ / 2/3*a\\
|-1 + cot |------------|| |-1 + cot |---||
\ \2*(-2 + 3*a)// \ \ 2 //
-------------------------- - -----------------
2 2
/ 2/ pi \\ / 2/3*a\\
|1 + cot |------------|| |1 + cot |---||
\ \2*(-2 + 3*a)// \ \ 2 //
$$- \frac{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{\pi}{2 \left(3 a - 2\right)} \right)} + 1\right)^{2}}$$
2
/pi \ / 2/ pi \ \
sin|-- + 6*a| | sin |-------| |
1 \2 / | \2 - 3*a/ | 4/ pi \
- - - ------------- + |-1 + -------------------| *sin |--------|
2 2 | 4/ pi \| \-4 + 6*a/
| 4*sin |-----------||
\ \2*(2 - 3*a)//
$$\left(-1 + \frac{\sin^{2}{\left(\frac{\pi}{2 - 3 a} \right)}}{4 \sin^{4}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)}}\right)^{2} \sin^{4}{\left(\frac{\pi}{6 a - 4} \right)} - \frac{\sin{\left(6 a + \frac{\pi}{2} \right)}}{2} - \frac{1}{2}$$
2
1 + cos(6*a) / 2/ pi \\ 4/ pi \
- ------------ + |-1 + cot |-----------|| *sin |--------|
2 \ \2*(2 - 3*a)// \-4 + 6*a/
$$- \frac{\cos{\left(6 a \right)} + 1}{2} + \left(\cot^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{\pi}{6 a - 4} \right)}$$
2/ pi \ 2
cos |-------| - cos (3*a)
\2 - 3*a/
$$- \cos^{2}{\left(3 a \right)} + \cos^{2}{\left(\frac{\pi}{2 - 3 a} \right)}$$
2/pi pi \ 2/pi \
sin |-- + --------| - sin |-- + 3*a|
\2 -2 + 3*a/ \2 /
$$\sin^{2}{\left(\frac{\pi}{2} + \frac{\pi}{3 a - 2} \right)} - \sin^{2}{\left(3 a + \frac{\pi}{2} \right)}$$
1 1
-------------- - ---------
2/ pi \ 2
sec |--------| sec (3*a)
\-2 + 3*a/
$$\frac{1}{\sec^{2}{\left(\frac{\pi}{3 a - 2} \right)}} - \frac{1}{\sec^{2}{\left(3 a \right)}}$$
1 1
------------------ - --------------
2/pi pi \ 2/pi \
csc |-- - -------| csc |-- - 3*a|
\2 2 - 3*a/ \2 /
$$- \frac{1}{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(\frac{\pi}{2} - \frac{\pi}{2 - 3 a} \right)}}$$
2
/ 2/ pi pi \\
| sec |- -- + -----------||
| \ 2 2*(2 - 3*a)/|
|-1 + ------------------------|
| 2/ pi \ |
| sec |-----------| |
1 1 \ \2*(2 - 3*a)/ /
- - - ---------- + --------------------------------
2 2*sec(6*a) 4/ pi pi \
sec |- -- + --------|
\ 2 -4 + 6*a/
$$\frac{\left(-1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(2 - 3 a\right)} \right)}}{\sec^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)}}\right)^{2}}{\sec^{4}{\left(- \frac{\pi}{2} + \frac{\pi}{6 a - 4} \right)}} - \frac{1}{2} - \frac{1}{2 \sec{\left(6 a \right)}}$$
2 2
/ 2/ pi \\ / 2/3*a\\
|-1 + cot |-----------|| |-1 + cot |---||
\ \2*(2 - 3*a)// \ \ 2 //
------------------------- - -----------------
2 2
/ 2/ pi \\ / 2/3*a\\
|1 + cot |-----------|| |1 + cot |---||
\ \2*(2 - 3*a)// \ \ 2 //
$$- \frac{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)} + 1\right)^{2}}$$
2
1 cos(6*a) / 1 \ 8/ pi \ 4/ pi \
- - - -------- + 16*|-1 + --------------| *cos |---------|*tan |---------|
2 2 | 2/ pi \| \-8 + 12*a/ \-8 + 12*a/
| tan |--------||
\ \-4 + 6*a//
$$16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{\pi}{6 a - 4} \right)}}\right)^{2} \cos^{8}{\left(\frac{\pi}{12 a - 8} \right)} \tan^{4}{\left(\frac{\pi}{12 a - 8} \right)} - \frac{\cos{\left(6 a \right)}}{2} - \frac{1}{2}$$
2
/ 1 \ 4/ pi \
16*|-1 + -----------------| *tan |------------|
| 2/ pi \| \2*(-4 + 6*a)/
2 | tan |-----------||
1 1 - tan (3*a) \ \2*(2 - 3*a)//
- - - ----------------- + -----------------------------------------------
2 / 2 \ 4
2*\1 + tan (3*a)/ / 2/ pi \\
|1 + tan |------------||
\ \2*(-4 + 6*a)//
$$\frac{16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)}}\right)^{2} \tan^{4}{\left(\frac{\pi}{2 \left(6 a - 4\right)} \right)}}{\left(\tan^{2}{\left(\frac{\pi}{2 \left(6 a - 4\right)} \right)} + 1\right)^{4}} - \frac{1 - \tan^{2}{\left(3 a \right)}}{2 \left(\tan^{2}{\left(3 a \right)} + 1\right)} - \frac{1}{2}$$
2 1 cos(6*a) / 2/ pi \\ 4/ pi \ 8/ pi \ - - - -------- + 16*|-1 + cot |-----------|| *cot |------------|*sin |---------| 2 2 \ \2*(2 - 3*a)// \2*(-4 + 6*a)/ \-8 + 12*a/
$$16 \left(\cot^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)} - 1\right)^{2} \sin^{8}{\left(\frac{\pi}{12 a - 8} \right)} \cot^{4}{\left(\frac{\pi}{2 \left(6 a - 4\right)} \right)} - \frac{\cos{\left(6 a \right)}}{2} - \frac{1}{2}$$
2 2
/ 2/ pi \\ / 2/3*a\\
|1 - tan |-----------|| |1 - tan |---||
\ \2*(2 - 3*a)// \ \ 2 //
------------------------ - ----------------
2 2
/ 2/ pi \\ / 2/3*a\\
|1 + tan |-----------|| |1 + tan |---||
\ \2*(2 - 3*a)// \ \ 2 //
$$- \frac{\left(1 - \tan^{2}{\left(\frac{3 a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}} + \frac{\left(1 - \tan^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{\pi}{2 \left(2 - 3 a\right)} \right)} + 1\right)^{2}}$$
(1 - tan(pi/(2*(2 - 3*a)))^2)^2/(1 + tan(pi/(2*(2 - 3*a)))^2)^2 - (1 - tan(3*a/2)^2)^2/(1 + tan(3*a/2)^2)^2