Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(a)^(2)/(1+tan(a)^(2))+cos(a)^(2)/(1+cot(a)^(2))
¿Cómo vas a descomponer esta sin(a)^(2)/(1+tan(a)^(2))+cos(a)^(2)/(1+cot(a)^(2)) expresión en fracciones?
Solución
Ha introducido
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2 2
sin (a) cos (a)
----------- + -----------
2 2
1 + tan (a) 1 + cot (a)
$$\frac{\cos^{2}{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} + \frac{\sin^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
sin(a)^2/(1 + tan(a)^2) + cos(a)^2/(1 + cot(a)^2)
Simplificación general
[src]
1 cos(4*a) - - -------- 4 4
$$\frac{1}{4} - \frac{\cos{\left(4 a \right)}}{4}$$
1/4 - cos(4*a)/4
Respuesta numérica
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cos(a)^2/(1.0 + cot(a)^2) + sin(a)^2/(1.0 + tan(a)^2)
cos(a)^2/(1.0 + cot(a)^2) + sin(a)^2/(1.0 + tan(a)^2)
Denominador racional
[src]
2 / 2 \ 2 / 2 \
cos (a)*\1 + tan (a)/ + sin (a)*\1 + cot (a)/
---------------------------------------------
/ 2 \ / 2 \
\1 + cot (a)/*\1 + tan (a)/
$$\frac{\left(\tan^{2}{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)} + \left(\cot^{2}{\left(a \right)} + 1\right) \sin^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
(cos(a)^2*(1 + tan(a)^2) + sin(a)^2*(1 + cot(a)^2))/((1 + cot(a)^2)*(1 + tan(a)^2))
Combinatoria
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2 2 2 2 2 2
cos (a) + sin (a) + cos (a)*tan (a) + cot (a)*sin (a)
-----------------------------------------------------
/ 2 \ / 2 \
\1 + cot (a)/*\1 + tan (a)/
$$\frac{\sin^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
(cos(a)^2 + sin(a)^2 + cos(a)^2*tan(a)^2 + cot(a)^2*sin(a)^2)/((1 + cot(a)^2)*(1 + tan(a)^2))
Potencias
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2
/ I*a -I*a\
|e e | 2
|---- + -----| / -I*a I*a\
\ 2 2 / \- e + e /
--------------- - -------------------------
2 / 2\
1 + cot (a) | / I*a -I*a\ |
| \- e + e / |
4*|1 - -----------------|
| 2 |
| / I*a -I*a\ |
\ \e + e / /
$$\frac{\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}}{\cot^{2}{\left(a \right)} + 1} - \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4 \left(- \frac{\left(- e^{i a} + e^{- i a}\right)^{2}}{\left(e^{i a} + e^{- i a}\right)^{2}} + 1\right)}$$
(exp(i*a)/2 + exp(-i*a)/2)^2/(1 + cot(a)^2) - (-exp(-i*a) + exp(i*a))^2/(4*(1 - (-exp(i*a) + exp(-i*a))^2/(exp(i*a) + exp(-i*a))^2))
Unión de expresiones racionales
[src]
2 / 2 \ 2 / 2 \
cos (a)*\1 + tan (a)/ + sin (a)*\1 + cot (a)/
---------------------------------------------
/ 2 \ / 2 \
\1 + cot (a)/*\1 + tan (a)/
$$\frac{\left(\tan^{2}{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)} + \left(\cot^{2}{\left(a \right)} + 1\right) \sin^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
(cos(a)^2*(1 + tan(a)^2) + sin(a)^2*(1 + cot(a)^2))/((1 + cot(a)^2)*(1 + tan(a)^2))
Denominador común
[src]
2 2 2 2 2 2
cos (a) + sin (a) + cos (a)*tan (a) + cot (a)*sin (a)
-----------------------------------------------------
2 2 2 2
1 + cot (a) + tan (a) + cot (a)*tan (a)
$$\frac{\sin^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \tan^{2}{\left(a \right)} + \cot^{2}{\left(a \right)} + 1}$$
(cos(a)^2 + sin(a)^2 + cos(a)^2*tan(a)^2 + cot(a)^2*sin(a)^2)/(1 + cot(a)^2 + tan(a)^2 + cot(a)^2*tan(a)^2)
Parte trigonométrica
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1 1 -------------------------- + ------------------------------- / 2/pi \\ / 2 \ | csc |-- - a|| | csc (a) | 2/pi \ | \2 /| 2 |1 + ------------|*csc |-- - a| |1 + ------------|*csc (a) | 2/pi \| \2 / | 2 | | csc |-- - a|| \ csc (a) / \ \2 //
$$\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$
2 / 2 2/a\ 4/a\\
sin (a)*|2 - sin (a) - 4*sin |-| + 4*sin |-||
\ \2/ \2//
$$\left(4 \sin^{4}{\left(\frac{a}{2} \right)} - 4 \sin^{2}{\left(\frac{a}{2} \right)} - \sin^{2}{\left(a \right)} + 2\right) \sin^{2}{\left(a \right)}$$
2/ pi\
cos |a - --| 2
\ 2 / cos (a)
---------------- + ----------------
2/ pi\ 2
cos |a - --| cos (a)
\ 2 / 1 + ------------
1 + ------------ 2/ pi\
2 cos |a - --|
cos (a) \ 2 /
$$\frac{\cos^{2}{\left(a \right)}}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}$$
/ 2/a\ 2/a\ 4/a\ \
| 16*cot |-| 4*cot |-| 64*cot |-| |
2/a\ | \4/ \2/ \4/ |
4*cot |-|*|2 - -------------- - -------------- + --------------|
\2/ | 2 2 4|
| / 2/a\\ / 2/a\\ / 2/a\\ |
| |1 + cot |-|| |1 + cot |-|| |1 + cot |-|| |
\ \ \4// \ \2// \ \4// /
----------------------------------------------------------------
2
/ 2/a\\
|1 + cot |-||
\ \2//
$$\frac{4 \left(2 - \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{16 \cot^{2}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{64 \cot^{4}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}}\right) \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2
/ 2/a\\ 2/a\
|-1 + cot |-|| 4*cot |-|
\ \2// \2/
---------------------------- + ----------------------------
2 2
/ 2 \ / 2/a\\ / 1 \ / 2/a\\
\1 + cot (a)/*|1 + cot |-|| |1 + -------|*|1 + cot |-||
\ \2// | 2 | \ \2//
\ cot (a)/
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(a \right)} + 1\right)} + \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2
1 -1 + cot (2*a)
- - -----------------
4 / 2 \
4*\1 + cot (2*a)/
$$- \frac{\cot^{2}{\left(2 a \right)} - 1}{4 \left(\cot^{2}{\left(2 a \right)} + 1\right)} + \frac{1}{4}$$
2/ pi\ / 2/ pi\ 2/a pi\ 4/a pi\\
cos |a - --|*|2 - cos |a - --| - 4*cos |- - --| + 4*cos |- - --||
\ 2 / \ \ 2 / \2 2 / \2 2 //
$$\left(4 \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} - 4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} - \cos^{2}{\left(a - \frac{\pi}{2} \right)} + 2\right) \cos^{2}{\left(a - \frac{\pi}{2} \right)}$$
4/a\ 2/a\ / 1 - cos(2*a) 8/a\ 4/a\\
4*cos |-|*tan |-|*|2 - 2*(1 - cos(a)) - ------------ + 64*cos |-|*tan |-||
\2/ \2/ \ 2 \4/ \4//
$$4 \left(- 2 \left(1 - \cos{\left(a \right)}\right) - \frac{1 - \cos{\left(2 a \right)}}{2} + 64 \cos^{8}{\left(\frac{a}{4} \right)} \tan^{4}{\left(\frac{a}{4} \right)} + 2\right) \cos^{4}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)}$$
1 4 4
2 - ------------ - ------------ + ------------
2/ pi\ 2/a pi\ 4/a pi\
sec |a - --| sec |- - --| sec |- - --|
\ 2 / \2 2 / \2 2 /
----------------------------------------------
2/ pi\
sec |a - --|
\ 2 /
$$\frac{2 - \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - \frac{4}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{4}{\sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
/ 2/a\ 2/a\ 4/a\ \
| 16*tan |-| 4*tan |-| 64*tan |-| |
2/a\ | \4/ \2/ \4/ |
4*tan |-|*|2 - -------------- - -------------- + --------------|
\2/ | 2 2 4|
| / 2/a\\ / 2/a\\ / 2/a\\ |
| |1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |
\ \ \4// \ \2// \ \4// /
----------------------------------------------------------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{4 \left(2 - \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{64 \tan^{4}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}}\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2
/ 2/a\\ 2/a\
|1 - tan |-|| 4*tan |-|
\ \2// \2/
---------------------------- + ----------------------------
2 2
/ 1 \ / 2/a\\ / 2 \ / 2/a\\
|1 + -------|*|1 + tan |-|| \1 + tan (a)/*|1 + tan |-||
| 2 | \ \2// \ \2//
\ tan (a)/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(a \right)} + 1\right)}$$
2 2
sin (a) cos (a)
------------- + -------------
4 2
4*sin (a) sin (2*a)
1 + --------- 1 + ---------
2 4
sin (2*a) 4*sin (a)
$$\frac{\sin^{2}{\left(a \right)}}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + \frac{\cos^{2}{\left(a \right)}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}$$
1 1 --------------------- + --------------------- / 2 \ / 2 \ | sec (a)| 2 | csc (a)| 2 |1 + -------|*csc (a) |1 + -------|*sec (a) | 2 | | 2 | \ csc (a)/ \ sec (a)/
$$\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} + 1\right) \sec^{2}{\left(a \right)}} + \frac{1}{\left(1 + \frac{\sec^{2}{\left(a \right)}}{\csc^{2}{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$
2/a\ 4/a\ / 1 - cos(2*a) 4/a\ 8/a\\
4*cot |-|*sin |-|*|2 - 2*(1 - cos(a)) - ------------ + 64*cot |-|*sin |-||
\2/ \2/ \ 2 \4/ \4//
$$4 \left(- 2 \left(1 - \cos{\left(a \right)}\right) - \frac{1 - \cos{\left(2 a \right)}}{2} + 64 \sin^{8}{\left(\frac{a}{4} \right)} \cot^{4}{\left(\frac{a}{4} \right)} + 2\right) \sin^{4}{\left(\frac{a}{2} \right)} \cot^{2}{\left(\frac{a}{2} \right)}$$
1 cos(4*a) - - -------- 4 4
$$\frac{1}{4} - \frac{\cos{\left(4 a \right)}}{4}$$
1 1
- - ---------------
4 /pi \
4*csc|-- - 4*a|
\2 /
$$\frac{1}{4} - \frac{1}{4 \csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$
1 1 - - ---------- 4 4*sec(4*a)
$$\frac{1}{4} - \frac{1}{4 \sec{\left(4 a \right)}}$$
/pi \
sin|-- + 4*a|
1 \2 /
- - -------------
4 4
$$\frac{1}{4} - \frac{\sin{\left(4 a + \frac{\pi}{2} \right)}}{4}$$
2/ pi\
2 sin |a + --|
sin (a) \ 2 /
------------- + -------------
4 2
4*sin (a) sin (2*a)
1 + --------- 1 + ---------
2 4
sin (2*a) 4*sin (a)
$$\frac{\sin^{2}{\left(a \right)}}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + \frac{\sin^{2}{\left(a + \frac{\pi}{2} \right)}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}$$
1 1 -------------------------- + ------------------------------- / 2/ pi\\ / 2 \ | sec |a - --|| | sec (a) | 2/ pi\ | \ 2 /| 2 |1 + ------------|*sec |a - --| |1 + ------------|*sec (a) | 2/ pi\| \ 2 / | 2 | | sec |a - --|| \ sec (a) / \ \ 2 //
$$\frac{1}{\left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec^{2}{\left(a \right)}}$$
2
1 1 - tan (2*a)
- - -----------------
4 / 2 \
4*\1 + tan (2*a)/
$$- \frac{1 - \tan^{2}{\left(2 a \right)}}{4 \left(\tan^{2}{\left(2 a \right)} + 1\right)} + \frac{1}{4}$$
2 2
sin (a) cos (a)
----------- + -----------
2 2
sin (a) cos (a)
1 + ------- 1 + -------
2 2
cos (a) sin (a)
$$\frac{\sin^{2}{\left(a \right)}}{\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(a \right)}} + 1} + \frac{\cos^{2}{\left(a \right)}}{1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}}$$
1 4 4
2 - ------- - ------- + -------
2 2/a\ 4/a\
csc (a) csc |-| csc |-|
\2/ \2/
-------------------------------
2
csc (a)
$$\frac{2 - \frac{1}{\csc^{2}{\left(a \right)}} - \frac{4}{\csc^{2}{\left(\frac{a}{2} \right)}} + \frac{4}{\csc^{4}{\left(\frac{a}{2} \right)}}}{\csc^{2}{\left(a \right)}}$$
(2 - 1/csc(a)^2 - 4/csc(a/2)^2 + 4/csc(a/2)^4)/csc(a)^2