Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
tg4a*ctg4a-((1-cos^2*9a)/(sin^2*9a-1))
¿Cómo vas a descomponer esta tg4a*ctg4a-((1-cos^2*9a)/(sin^2*9a-1)) expresión en fracciones?
Solución
Ha introducido
[src]
2
1 - cos (9)*a
tan(4*a)*cot(4*a) - -------------
2
sin (9)*a - 1
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} - \frac{- a \cos^{2}{\left(9 \right)} + 1}{a \sin^{2}{\left(9 \right)} - 1}$$
tan(4*a)*cot(4*a) - (1 - cos(9)^2*a)/(sin(9)^2*a - 1)
Simplificación general
[src]
-2 + a
--------------
2
-1 + a*sin (9)
$$\frac{a - 2}{a \sin^{2}{\left(9 \right)} - 1}$$
(-2 + a)/(-1 + a*sin(9)^2)
Respuesta numérica
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cot(4*a)*tan(4*a) - (1.0 - 0.83015835412204*a)/(-1.0 + 0.16984164587796*a)
cot(4*a)*tan(4*a) - (1.0 - 0.83015835412204*a)/(-1.0 + 0.16984164587796*a)
Denominador racional
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2 / 2 \
-1 + a*cos (9) + \-1 + a*sin (9)/*cot(4*a)*tan(4*a)
---------------------------------------------------
2
-1 + a*sin (9)
$$\frac{a \cos^{2}{\left(9 \right)} + \left(a \sin^{2}{\left(9 \right)} - 1\right) \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{a \sin^{2}{\left(9 \right)} - 1}$$
(-1 + a*cos(9)^2 + (-1 + a*sin(9)^2)*cot(4*a)*tan(4*a))/(-1 + a*sin(9)^2)
Unión de expresiones racionales
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2 / 2 \
-1 + a*cos (9) + \-1 + a*sin (9)/*cot(4*a)*tan(4*a)
---------------------------------------------------
2
-1 + a*sin (9)
$$\frac{a \cos^{2}{\left(9 \right)} + \left(a \sin^{2}{\left(9 \right)} - 1\right) \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{a \sin^{2}{\left(9 \right)} - 1}$$
(-1 + a*cos(9)^2 + (-1 + a*sin(9)^2)*cot(4*a)*tan(4*a))/(-1 + a*sin(9)^2)
Potencias
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2
-1 + a*cos (9)
-------------- + cot(4*a)*tan(4*a)
2
-1 + a*sin (9)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} + \frac{a \cos^{2}{\left(9 \right)} - 1}{a \sin^{2}{\left(9 \right)} - 1}$$
2
/ -9*I 9*I\
|e e |
1 - a*|----- + ----| / 4*I*a -4*I*a\
\ 2 2 / I*\- e + e /*cot(4*a)
- ------------------------ + -------------------------------
2 -4*I*a 4*I*a
/ -9*I 9*I\ e + e
a*\- e + e /
-1 - -------------------
4
$$- \frac{- a \left(\frac{e^{- 9 i}}{2} + \frac{e^{9 i}}{2}\right)^{2} + 1}{- \frac{a \left(e^{9 i} - e^{- 9 i}\right)^{2}}{4} - 1} + \frac{i \left(- e^{4 i a} + e^{- 4 i a}\right) \cot{\left(4 a \right)}}{e^{4 i a} + e^{- 4 i a}}$$
-(1 - a*(exp(-9*i)/2 + exp(9*i)/2)^2)/(-1 - a*(-exp(-9*i) + exp(9*i))^2/4) + i*(-exp(4*i*a) + exp(-4*i*a))*cot(4*a)/(exp(-4*i*a) + exp(4*i*a))
Combinatoria
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2 2
-1 + a*cos (9) - cot(4*a)*tan(4*a) + a*sin (9)*cot(4*a)*tan(4*a)
----------------------------------------------------------------
2
-1 + a*sin (9)
$$\frac{a \sin^{2}{\left(9 \right)} \tan{\left(4 a \right)} \cot{\left(4 a \right)} + a \cos^{2}{\left(9 \right)} - \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{a \sin^{2}{\left(9 \right)} - 1}$$
(-1 + a*cos(9)^2 - cot(4*a)*tan(4*a) + a*sin(9)^2*cot(4*a)*tan(4*a))/(-1 + a*sin(9)^2)
Denominador común
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2
-1 + a*cos (9)
-------------- + cot(4*a)*tan(4*a)
2
-1 + a*sin (9)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} + \frac{a \cos^{2}{\left(9 \right)} - 1}{a \sin^{2}{\left(9 \right)} - 1}$$
(-1 + a*cos(9)^2)/(-1 + a*sin(9)^2) + cot(4*a)*tan(4*a)
Parte trigonométrica
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2
/ 2/ 9 pi\\
| csc |- - + --||
| \ 2 2 /|
|1 + --------------| *(-2 + a)
| 2 |
\ csc (9/2) /
-----------------------------------------------------------
4/ 9 pi\ 2/ 9 pi\ 2/ 9 pi\
csc |- - + --| 2*csc |- - + --| 4*a*csc |- - + --|
\ 2 2 / \ 2 2 / \ 2 2 /
-1 - -------------- - ---------------- + ------------------
4 2 2
csc (9/2) csc (9/2) csc (9/2)
$$\frac{\left(1 + \frac{\csc^{2}{\left(- \frac{9}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{9}{2} \right)}}\right)^{2} \left(a - 2\right)}{\frac{4 a \csc^{2}{\left(- \frac{9}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{9}{2} \right)}} - \frac{\csc^{4}{\left(- \frac{9}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{9}{2} \right)}} - \frac{2 \csc^{2}{\left(- \frac{9}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{9}{2} \right)}} - 1}$$
2
1 - a*cos (9)
1 - --------------
2
-1 + a*sin (9)
$$1 - \frac{- a \cos^{2}{\left(9 \right)} + 1}{a \sin^{2}{\left(9 \right)} - 1}$$
a
1 - -------
2
sec (9)
1 - -----------------
a
-1 + ------------
2/ pi\
sec |9 - --|
\ 2 /
$$- \frac{- \frac{a}{\sec^{2}{\left(9 \right)}} + 1}{\frac{a}{\sec^{2}{\left(9 - \frac{\pi}{2} \right)}} - 1} + 1$$
2
/ 2 \
\1 + tan (9/2)/ *(-2 + a)
--------------------------------------------
4 2 2
-1 - tan (9/2) - 2*tan (9/2) + 4*a*tan (9/2)
$$\frac{\left(1 + \tan^{2}{\left(\frac{9}{2} \right)}\right)^{2} \left(a - 2\right)}{4 a \tan^{2}{\left(\frac{9}{2} \right)} - \tan^{4}{\left(\frac{9}{2} \right)} - 2 \tan^{2}{\left(\frac{9}{2} \right)} - 1}$$
2
/ 4 \
| 4*sin (9/2)|
|1 + -----------| *(-2 + a)
| 2 |
\ sin (9) /
---------------------------------------------
4 4
4 8*sin (9/2) 16*a*sin (9/2)
-1 - tan (9/2) - ----------- + --------------
2 2
sin (9) sin (9)
$$\frac{\left(1 + \frac{4 \sin^{4}{\left(\frac{9}{2} \right)}}{\sin^{2}{\left(9 \right)}}\right)^{2} \left(a - 2\right)}{\frac{16 a \sin^{4}{\left(\frac{9}{2} \right)}}{\sin^{2}{\left(9 \right)}} - \tan^{4}{\left(\frac{9}{2} \right)} - \frac{8 \sin^{4}{\left(\frac{9}{2} \right)}}{\sin^{2}{\left(9 \right)}} - 1}$$
2*(2 - a)
----------------------
/ pi\
2 - a + a*sin|18 + --|
\ 2 /
$$\frac{2 \left(2 - a\right)}{- a + a \sin{\left(\frac{\pi}{2} + 18 \right)} + 2}$$
-2 + a
-----------------
a
-1 + ------------
2/ pi\
sec |9 - --|
\ 2 /
$$\frac{a - 2}{\frac{a}{\sec^{2}{\left(9 - \frac{\pi}{2} \right)}} - 1}$$
4 - 2*a ----------------- 2 - a + a*cos(18)
$$\frac{4 - 2 a}{- a + a \cos{\left(18 \right)} + 2}$$
2*(2 - a)
------------------------
/ 2 \
a*\-1 + cot (9)/
2 - a + ----------------
2
1 + cot (9)
$$\frac{2 \left(2 - a\right)}{- a + \frac{a \left(-1 + \cot^{2}{\left(9 \right)}\right)}{1 + \cot^{2}{\left(9 \right)}} + 2}$$
2
1 - a*cos (9)
1 - -------------------
2/ pi\
-1 + a*cos |9 - --|
\ 2 /
$$- \frac{- a \cos^{2}{\left(9 \right)} + 1}{a \cos^{2}{\left(9 - \frac{\pi}{2} \right)} - 1} + 1$$
-2 + a
---------------------
2
4*a*cot (9/2)
-1 + ----------------
2
/ 2 \
\1 + cot (9/2)/
$$\frac{a - 2}{\frac{4 a \cot^{2}{\left(\frac{9}{2} \right)}}{\left(\cot^{2}{\left(\frac{9}{2} \right)} + 1\right)^{2}} - 1}$$
-2 + a
---------------------
2
4*a*tan (9/2)
-1 + ----------------
2
/ 2 \
\1 + tan (9/2)/
$$\frac{a - 2}{\frac{4 a \tan^{2}{\left(\frac{9}{2} \right)}}{\left(1 + \tan^{2}{\left(\frac{9}{2} \right)}\right)^{2}} - 1}$$
2
/ 2 \
| sec (9/2) |
|1 + ------------| *(-2 + a)
| 2/9 pi\|
| sec |- - --||
\ \2 2 //
------------------------------------------------
4 2 2
sec (9/2) 2*sec (9/2) 4*a*sec (9/2)
-1 - ------------ - ------------ + -------------
4/9 pi\ 2/9 pi\ 2/9 pi\
sec |- - --| sec |- - --| sec |- - --|
\2 2 / \2 2 / \2 2 /
$$\frac{\left(1 + \frac{\sec^{2}{\left(\frac{9}{2} \right)}}{\sec^{2}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}}\right)^{2} \left(a - 2\right)}{\frac{4 a \sec^{2}{\left(\frac{9}{2} \right)}}{\sec^{2}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}} - \frac{\sec^{4}{\left(\frac{9}{2} \right)}}{\sec^{4}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}} - \frac{2 \sec^{2}{\left(\frac{9}{2} \right)}}{\sec^{2}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}} - 1}$$
2*(2 - a) ----------------- 2 - a + a*cos(18)
$$\frac{2 \left(2 - a\right)}{- a + a \cos{\left(18 \right)} + 2}$$
2
/ 2 \
a*\1 - tan (9/2)/
1 - ------------------
2
/ 2 \
\1 + tan (9/2)/
1 - ----------------------
2
4*a*tan (9/2)
-1 + ----------------
2
/ 2 \
\1 + tan (9/2)/
$$- \frac{- \frac{a \left(1 - \tan^{2}{\left(\frac{9}{2} \right)}\right)^{2}}{\left(1 + \tan^{2}{\left(\frac{9}{2} \right)}\right)^{2}} + 1}{\frac{4 a \tan^{2}{\left(\frac{9}{2} \right)}}{\left(1 + \tan^{2}{\left(\frac{9}{2} \right)}\right)^{2}} - 1} + 1$$
-8 + 4*a
------------------------------------------------------------
/ 2 2 \
2 | 4 4*sin (9/2) 8*a*sin (9/2)|
(1 + cos(9)) *|-1 - tan (9/2) - ----------- + -------------|
\ 1 + cos(9) 1 + cos(9) /
$$\frac{4 a - 8}{\left(\cos{\left(9 \right)} + 1\right)^{2} \left(\frac{8 a \sin^{2}{\left(\frac{9}{2} \right)}}{\cos{\left(9 \right)} + 1} - \tan^{4}{\left(\frac{9}{2} \right)} - \frac{4 \sin^{2}{\left(\frac{9}{2} \right)}}{\cos{\left(9 \right)} + 1} - 1\right)}$$
-2 + a
------------
a
-1 + -------
2
csc (9)
$$\frac{a - 2}{\frac{a}{\csc^{2}{\left(9 \right)}} - 1}$$
a
1 - -------------
2/ pi\
csc |-9 + --|
\ 2 /
1 - -----------------
a
-1 + -------
2
csc (9)
$$1 - \frac{- \frac{a}{\csc^{2}{\left(-9 + \frac{\pi}{2} \right)}} + 1}{\frac{a}{\csc^{2}{\left(9 \right)}} - 1}$$
2
/ 4 \
| 4*sin (9/2)|
|1 + -----------| *(-2 + a)
| 2 |
\ sin (9) /
------------------------------------------------
8 4 4
16*sin (9/2) 8*sin (9/2) 16*a*sin (9/2)
-1 - ------------ - ----------- + --------------
4 2 2
sin (9) sin (9) sin (9)
$$\frac{\left(1 + \frac{4 \sin^{4}{\left(\frac{9}{2} \right)}}{\sin^{2}{\left(9 \right)}}\right)^{2} \left(a - 2\right)}{\frac{16 a \sin^{4}{\left(\frac{9}{2} \right)}}{\sin^{2}{\left(9 \right)}} - \frac{16 \sin^{8}{\left(\frac{9}{2} \right)}}{\sin^{4}{\left(9 \right)}} - \frac{8 \sin^{4}{\left(\frac{9}{2} \right)}}{\sin^{2}{\left(9 \right)}} - 1}$$
2
/ 2/9 pi\\
| cos |- - --||
| \2 2 /|
|1 + ------------| *(-2 + a)
| 2 |
\ cos (9/2) /
-----------------------------------------------------
4/9 pi\ 2/9 pi\ 2/9 pi\
cos |- - --| 2*cos |- - --| 4*a*cos |- - --|
\2 2 / \2 2 / \2 2 /
-1 - ------------ - -------------- + ----------------
4 2 2
cos (9/2) cos (9/2) cos (9/2)
$$\frac{\left(1 + \frac{\cos^{2}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{9}{2} \right)}}\right)^{2} \left(a - 2\right)}{\frac{4 a \cos^{2}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{9}{2} \right)}} - \frac{\cos^{4}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{9}{2} \right)}} - \frac{2 \cos^{2}{\left(\frac{9}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{9}{2} \right)}} - 1}$$
2*(2 - a)
-----------------------
/ 2 \
a*\1 - tan (9)/
2 - a + ---------------
2
1 + tan (9)
$$\frac{2 \left(2 - a\right)}{- a + \frac{a \left(1 - \tan^{2}{\left(9 \right)}\right)}{\tan^{2}{\left(9 \right)} + 1} + 2}$$
2
/ 2 \
a*\-1 + cot (9/2)/
1 - -------------------
2
/ 2 \
\1 + cot (9/2)/
1 - -----------------------
2
4*a*cot (9/2)
-1 + ----------------
2
/ 2 \
\1 + cot (9/2)/
$$- \frac{- \frac{a \left(-1 + \cot^{2}{\left(\frac{9}{2} \right)}\right)^{2}}{\left(\cot^{2}{\left(\frac{9}{2} \right)} + 1\right)^{2}} + 1}{\frac{4 a \cot^{2}{\left(\frac{9}{2} \right)}}{\left(\cot^{2}{\left(\frac{9}{2} \right)} + 1\right)^{2}} - 1} + 1$$
2*(2 - a)
---------------
a
2 - a + -------
sec(18)
$$\frac{2 \left(2 - a\right)}{- a + \frac{a}{\sec{\left(18 \right)}} + 2}$$
2*(2 - a)
---------------------
a
2 - a + -------------
/ pi\
csc|-18 + --|
\ 2 /
$$\frac{2 \left(2 - a\right)}{- a + \frac{a}{\csc{\left(-18 + \frac{\pi}{2} \right)}} + 2}$$
2/ pi\
1 - a*sin |9 + --|
\ 2 /
1 - ------------------
2
-1 + a*sin (9)
$$1 - \frac{- a \sin^{2}{\left(\frac{\pi}{2} + 9 \right)} + 1}{a \sin^{2}{\left(9 \right)} - 1}$$
-2 + a
--------------
2
-1 + a*sin (9)
$$\frac{a - 2}{a \sin^{2}{\left(9 \right)} - 1}$$
-2 + a
--------------------
a*(1 - cos(18))
-1 + ---------------
2
$$\frac{a - 2}{\frac{a \left(1 - \cos{\left(18 \right)}\right)}{2} - 1}$$
2
/ 1 \
|1 + ---------| *(-2 + a)
| 2 |
\ cot (9/2)/
--------------------------------------
1 2 4*a
-1 - --------- - --------- + ---------
4 2 2
cot (9/2) cot (9/2) cot (9/2)
$$\frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{9}{2} \right)}}\right)^{2} \left(a - 2\right)}{\frac{4 a}{\cot^{2}{\left(\frac{9}{2} \right)}} - \frac{1}{\cot^{4}{\left(\frac{9}{2} \right)}} - \frac{2}{\cot^{2}{\left(\frac{9}{2} \right)}} - 1}$$
a
1 - -------
2
sec (9)
1 - ------------
a
-1 + -------
2
csc (9)
$$1 - \frac{- \frac{a}{\sec^{2}{\left(9 \right)}} + 1}{\frac{a}{\csc^{2}{\left(9 \right)}} - 1}$$
-2 + a
-------------------
2/ pi\
-1 + a*cos |9 - --|
\ 2 /
$$\frac{a - 2}{a \cos^{2}{\left(9 - \frac{\pi}{2} \right)} - 1}$$
(-2 + a)/(-1 + a*cos(9 - pi/2)^2)