Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
tan(t)/(1-tan(t)^(2))+cot(t)/(1-cot(t)^(2))
¿Cómo vas a descomponer esta tan(t)/(1-tan(t)^(2))+cot(t)/(1-cot(t)^(2)) expresión en fracciones?
Solución
Ha introducido
[src]
tan(t) cot(t)
----------- + -----------
2 2
1 - tan (t) 1 - cot (t)
$$\frac{\cot{\left(t \right)}}{1 - \cot^{2}{\left(t \right)}} + \frac{\tan{\left(t \right)}}{1 - \tan^{2}{\left(t \right)}}$$
tan(t)/(1 - tan(t)^2) + cot(t)/(1 - cot(t)^2)
Respuesta numérica
[src]
cot(t)/(1.0 - cot(t)^2) + tan(t)/(1.0 - tan(t)^2)
cot(t)/(1.0 - cot(t)^2) + tan(t)/(1.0 - tan(t)^2)
Denominador común
[src]
/ 2 2 \
-\-cot(t) - tan(t) + cot (t)*tan(t) + tan (t)*cot(t)/
------------------------------------------------------
2 2 2 2
1 - cot (t) - tan (t) + cot (t)*tan (t)
$$- \frac{\tan^{2}{\left(t \right)} \cot{\left(t \right)} + \tan{\left(t \right)} \cot^{2}{\left(t \right)} - \tan{\left(t \right)} - \cot{\left(t \right)}}{\tan^{2}{\left(t \right)} \cot^{2}{\left(t \right)} - \tan^{2}{\left(t \right)} - \cot^{2}{\left(t \right)} + 1}$$
-(-cot(t) - tan(t) + cot(t)^2*tan(t) + tan(t)^2*cot(t))/(1 - cot(t)^2 - tan(t)^2 + cot(t)^2*tan(t)^2)
Potencias
[src]
/ I*t -I*t\
cot(t) I*\- e + e /
----------- + --------------------------------------
2 / 2\
1 - cot (t) | / I*t -I*t\ |
| \- e + e / | / I*t -I*t\
|1 + -----------------|*\e + e /
| 2 |
| / I*t -I*t\ |
\ \e + e / /
$$\frac{i \left(- e^{i t} + e^{- i t}\right)}{\left(\frac{\left(- e^{i t} + e^{- i t}\right)^{2}}{\left(e^{i t} + e^{- i t}\right)^{2}} + 1\right) \left(e^{i t} + e^{- i t}\right)} + \frac{\cot{\left(t \right)}}{1 - \cot^{2}{\left(t \right)}}$$
cot(t)/(1 - cot(t)^2) + i*(-exp(i*t) + exp(-i*t))/((1 + (-exp(i*t) + exp(-i*t))^2/(exp(i*t) + exp(-i*t))^2)*(exp(i*t) + exp(-i*t)))
Unión de expresiones racionales
[src]
/ 2 \ / 2 \
\1 - cot (t)/*tan(t) + \1 - tan (t)/*cot(t)
-------------------------------------------
/ 2 \ / 2 \
\1 - cot (t)/*\1 - tan (t)/
$$\frac{\left(1 - \tan^{2}{\left(t \right)}\right) \cot{\left(t \right)} + \left(1 - \cot^{2}{\left(t \right)}\right) \tan{\left(t \right)}}{\left(1 - \tan^{2}{\left(t \right)}\right) \left(1 - \cot^{2}{\left(t \right)}\right)}$$
((1 - cot(t)^2)*tan(t) + (1 - tan(t)^2)*cot(t))/((1 - cot(t)^2)*(1 - tan(t)^2))
Combinatoria
[src]
-(-1 + cot(t)*tan(t))*(cot(t) + tan(t)) ----------------------------------------------------- (1 + cot(t))*(1 + tan(t))*(-1 + cot(t))*(-1 + tan(t))
$$- \frac{\left(\tan{\left(t \right)} \cot{\left(t \right)} - 1\right) \left(\tan{\left(t \right)} + \cot{\left(t \right)}\right)}{\left(\tan{\left(t \right)} - 1\right) \left(\tan{\left(t \right)} + 1\right) \left(\cot{\left(t \right)} - 1\right) \left(\cot{\left(t \right)} + 1\right)}$$
-(-1 + cot(t)*tan(t))*(cot(t) + tan(t))/((1 + cot(t))*(1 + tan(t))*(-1 + cot(t))*(-1 + tan(t)))
Parte trigonométrica
[src]
/pi \
csc|-- - 2*t|
\2 / 2*csc(2*t)
- ------------- + --------------------------
2*csc(2*t) / 2/pi \\
| csc |-- - t||
| \2 /| 2
|1 - ------------|*csc (t)
| 2 |
\ csc (t) /
$$- \frac{\csc{\left(- 2 t + \frac{\pi}{2} \right)}}{2 \csc{\left(2 t \right)}} + \frac{2 \csc{\left(2 t \right)}}{\left(1 - \frac{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}\right) \csc^{2}{\left(t \right)}}$$
2
sin(2*t) 2*sin (t)
------------------------- + ------------------------
/ 2 \ / 4 \
| sin (2*t)| 2 | 4*sin (t)|
2*|1 - ---------|*sin (t) |1 - ---------|*sin(2*t)
| 4 | | 2 |
\ 4*sin (t)/ \ sin (2*t)/
$$\frac{2 \sin^{2}{\left(t \right)}}{\left(- \frac{4 \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}} + 1\right) \sin{\left(2 t \right)}} + \frac{\sin{\left(2 t \right)}}{2 \left(1 - \frac{\sin^{2}{\left(2 t \right)}}{4 \sin^{4}{\left(t \right)}}\right) \sin^{2}{\left(t \right)}}$$
1 tan(t) -------------------- + ----------- / 1 \ 2 |1 - -------|*tan(t) 1 - tan (t) | 2 | \ tan (t)/
$$\frac{\tan{\left(t \right)}}{1 - \tan^{2}{\left(t \right)}} + \frac{1}{\left(1 - \frac{1}{\tan^{2}{\left(t \right)}}\right) \tan{\left(t \right)}}$$
2
tan(2*t) 2*sin (t)*csc(2*t)
- -------- + ------------------
2 2
1 - tan (t)
$$- \frac{\tan{\left(2 t \right)}}{2} + \frac{2 \sin^{2}{\left(t \right)} \csc{\left(2 t \right)}}{1 - \tan^{2}{\left(t \right)}}$$
1 cot(t) -------------------- + ----------- / 1 \ 2 |1 - -------|*cot(t) 1 - cot (t) | 2 | \ cot (t)/
$$\frac{\cot{\left(t \right)}}{1 - \cot^{2}{\left(t \right)}} + \frac{1}{\left(1 - \frac{1}{\cot^{2}{\left(t \right)}}\right) \cot{\left(t \right)}}$$
2
tan(2*t) 2*sin (t)
- -------- + ----------------------
2 / 2 \
\1 - tan (t)/*sin(2*t)
$$- \frac{\tan{\left(2 t \right)}}{2} + \frac{2 \sin^{2}{\left(t \right)}}{\left(1 - \tan^{2}{\left(t \right)}\right) \sin{\left(2 t \right)}}$$
/ pi\
2*sec|2*t - --|
sec(2*t) \ 2 /
- --------------- + -------------------------------
/ pi\ / 2 \
2*sec|2*t - --| | sec (t) | 2/ pi\
\ 2 / |1 - ------------|*sec |t - --|
| 2/ pi\| \ 2 /
| sec |t - --||
\ \ 2 //
$$- \frac{\sec{\left(2 t \right)}}{2 \sec{\left(2 t - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(2 t - \frac{\pi}{2} \right)}}{\left(- \frac{\sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
sin(t) cos(t) -------------------- + -------------------- / 2 \ / 2 \ | sin (t)| | cos (t)| |1 - -------|*cos(t) |1 - -------|*sin(t) | 2 | | 2 | \ cos (t)/ \ sin (t)/
$$\frac{\sin{\left(t \right)}}{\left(- \frac{\sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}} + 1\right) \cos{\left(t \right)}} + \frac{\cos{\left(t \right)}}{\left(1 - \frac{\cos^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}\right) \sin{\left(t \right)}}$$
2/t\ / 2 \
4*cot |-|*\1 + cot (t)/
1 \2/
- ---------- + -----------------------------------
2*cot(2*t) 2
/ 2/t\\ / 1 \
|1 + cot |-|| *|1 - -------|*cot(t)
\ \2// | 2 |
\ cot (t)/
$$- \frac{1}{2 \cot{\left(2 t \right)}} + \frac{4 \left(\cot^{2}{\left(t \right)} + 1\right) \cot^{2}{\left(\frac{t}{2} \right)}}{\left(1 - \frac{1}{\cot^{2}{\left(t \right)}}\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cot{\left(t \right)}}$$
2/t\ / 2 \
4*tan |-|*\1 + tan (t)/
tan(2*t) \2/
- -------- + -----------------------------------
2 2
/ 2/t\\ / 2 \
|1 + tan |-|| *\1 - tan (t)/*tan(t)
\ \2//
$$- \frac{\tan{\left(2 t \right)}}{2} + \frac{4 \left(\tan^{2}{\left(t \right)} + 1\right) \tan^{2}{\left(\frac{t}{2} \right)}}{\left(1 - \tan^{2}{\left(t \right)}\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan{\left(t \right)}}$$
2 2
sin (2*t) 2*sin (t)
- --------- + ------------------------
sin(4*t) / 4 \
| 4*sin (t)|
|1 - ---------|*sin(2*t)
| 2 |
\ sin (2*t)/
$$- \frac{\sin^{2}{\left(2 t \right)}}{\sin{\left(4 t \right)}} + \frac{2 \sin^{2}{\left(t \right)}}{\left(- \frac{4 \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}} + 1\right) \sin{\left(2 t \right)}}$$
/ pi\
sec|t - --|
\ 2 / sec(t)
------------------------- + ------------------------------
/ 2/ pi\\ / 2 \
| sec |t - --|| | sec (t) | / pi\
| \ 2 /| |1 - ------------|*sec|t - --|
|1 - ------------|*sec(t) | 2/ pi\| \ 2 /
| 2 | | sec |t - --||
\ sec (t) / \ \ 2 //
$$\frac{\sec{\left(t \right)}}{\left(- \frac{\sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(t - \frac{\pi}{2} \right)}} + \frac{\sec{\left(t - \frac{\pi}{2} \right)}}{\left(1 - \frac{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}{\sec^{2}{\left(t \right)}}\right) \sec{\left(t \right)}}$$
0
$$0$$
/ pi\ 2/ pi\
cos|2*t - --| 2*cos |t - --|
\ 2 / \ 2 /
- ------------- + --------------------------------
2*cos(2*t) / 2/ pi\\
| cos |t - --||
| \ 2 /| / pi\
|1 - ------------|*cos|2*t - --|
| 2 | \ 2 /
\ cos (t) /
$$- \frac{\cos{\left(2 t - \frac{\pi}{2} \right)}}{2 \cos{\left(2 t \right)}} + \frac{2 \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}\right) \cos{\left(2 t - \frac{\pi}{2} \right)}}$$
/pi \
csc|-- - t|
\2 / csc(t)
------------------------- + ------------------------------
/ 2/pi \\ / 2 \
| csc |-- - t|| | csc (t) | /pi \
| \2 /| |1 - ------------|*csc|-- - t|
|1 - ------------|*csc(t) | 2/pi \| \2 /
| 2 | | csc |-- - t||
\ csc (t) / \ \2 //
$$\frac{\csc{\left(t \right)}}{\left(- \frac{\csc^{2}{\left(t \right)}}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- t + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- t + \frac{\pi}{2} \right)}}{\left(1 - \frac{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}\right) \csc{\left(t \right)}}$$
sec(t) csc(t) -------------------- + -------------------- / 2 \ / 2 \ | sec (t)| | csc (t)| |1 - -------|*csc(t) |1 - -------|*sec(t) | 2 | | 2 | \ csc (t)/ \ sec (t)/
$$\frac{\csc{\left(t \right)}}{\left(- \frac{\csc^{2}{\left(t \right)}}{\sec^{2}{\left(t \right)}} + 1\right) \sec{\left(t \right)}} + \frac{\sec{\left(t \right)}}{\left(1 - \frac{\sec^{2}{\left(t \right)}}{\csc^{2}{\left(t \right)}}\right) \csc{\left(t \right)}}$$
/ pi\
cos|t - --|
\ 2 / cos(t)
------------------------- + ------------------------------
/ 2/ pi\\ / 2 \
| cos |t - --|| | cos (t) | / pi\
| \ 2 /| |1 - ------------|*cos|t - --|
|1 - ------------|*cos(t) | 2/ pi\| \ 2 /
| 2 | | cos |t - --||
\ cos (t) / \ \ 2 //
$$\frac{\cos{\left(t \right)}}{\left(- \frac{\cos^{2}{\left(t \right)}}{\cos^{2}{\left(t - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(t - \frac{\pi}{2} \right)}} + \frac{\cos{\left(t - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}\right) \cos{\left(t \right)}}$$
cos(t - pi/2)/((1 - cos(t - pi/2)^2/cos(t)^2)*cos(t)) + cos(t)/((1 - cos(t)^2/cos(t - pi/2)^2)*cos(t - pi/2))
Denominador racional
[src]
/ 2 \ / 2 \
\1 - cot (t)/*tan(t) + \1 - tan (t)/*cot(t)
-------------------------------------------
/ 2 \ / 2 \
\1 - cot (t)/*\1 - tan (t)/
$$\frac{\left(1 - \tan^{2}{\left(t \right)}\right) \cot{\left(t \right)} + \left(1 - \cot^{2}{\left(t \right)}\right) \tan{\left(t \right)}}{\left(1 - \tan^{2}{\left(t \right)}\right) \left(1 - \cot^{2}{\left(t \right)}\right)}$$
((1 - cot(t)^2)*tan(t) + (1 - tan(t)^2)*cot(t))/((1 - cot(t)^2)*(1 - tan(t)^2))