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