Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
tg(8a)*ctg(8a)-(cos^2*6a-1)/(1-sin^2(6a))
¿Cómo vas a descomponer esta tg(8a)*ctg(8a)-(cos^2*6a-1)/(1-sin^2(6a)) expresión en fracciones?
Solución
Ha introducido
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2
cos (6)*a - 1
tan(8*a)*cot(8*a) - -------------
2
1 - sin (6*a)
$$\tan{\left(8 a \right)} \cot{\left(8 a \right)} - \frac{a \cos^{2}{\left(6 \right)} - 1}{1 - \sin^{2}{\left(6 a \right)}}$$
tan(8*a)*cot(8*a) - (cos(6)^2*a - 1)/(1 - sin(6*a)^2)
Simplificación general
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3 cos(12*a) 2
- - - --------- + a*cos (6)
2 2
---------------------------
2
-1 + sin (6*a)
$$\frac{a \cos^{2}{\left(6 \right)} - \frac{\cos{\left(12 a \right)}}{2} - \frac{3}{2}}{\sin^{2}{\left(6 a \right)} - 1}$$
(-3/2 - cos(12*a)/2 + a*cos(6)^2)/(-1 + sin(6*a)^2)
Denominador racional
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2 / 2 \
1 - a*cos (6) + \1 - sin (6*a)/*cot(8*a)*tan(8*a)
-------------------------------------------------
2
1 - sin (6*a)
$$\frac{- a \cos^{2}{\left(6 \right)} + \left(1 - \sin^{2}{\left(6 a \right)}\right) \tan{\left(8 a \right)} \cot{\left(8 a \right)} + 1}{1 - \sin^{2}{\left(6 a \right)}}$$
(1 - a*cos(6)^2 + (1 - sin(6*a)^2)*cot(8*a)*tan(8*a))/(1 - sin(6*a)^2)
Respuesta numérica
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cot(8*a)*tan(8*a) - (-1.0 + 0.921926979366246*a)/(1.0 - sin(6*a)^2)
cot(8*a)*tan(8*a) - (-1.0 + 0.921926979366246*a)/(1.0 - sin(6*a)^2)
Unión de expresiones racionales
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2 / 2 \
1 - a*cos (6) + \1 - sin (6*a)/*cot(8*a)*tan(8*a)
-------------------------------------------------
2
1 - sin (6*a)
$$\frac{- a \cos^{2}{\left(6 \right)} + \left(1 - \sin^{2}{\left(6 a \right)}\right) \tan{\left(8 a \right)} \cot{\left(8 a \right)} + 1}{1 - \sin^{2}{\left(6 a \right)}}$$
(1 - a*cos(6)^2 + (1 - sin(6*a)^2)*cot(8*a)*tan(8*a))/(1 - sin(6*a)^2)
Denominador común
[src]
2
-1 + a*cos (6)
-------------- + cot(8*a)*tan(8*a)
2
-1 + sin (6*a)
$$\frac{a \cos^{2}{\left(6 \right)} - 1}{\sin^{2}{\left(6 a \right)} - 1} + \tan{\left(8 a \right)} \cot{\left(8 a \right)}$$
(-1 + a*cos(6)^2)/(-1 + sin(6*a)^2) + cot(8*a)*tan(8*a)
Potencias
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2
1 - a*cos (6)
------------- + cot(8*a)*tan(8*a)
2
1 - sin (6*a)
$$\tan{\left(8 a \right)} \cot{\left(8 a \right)} + \frac{- a \cos^{2}{\left(6 \right)} + 1}{1 - \sin^{2}{\left(6 a \right)}}$$
2
/ -6*I 6*I\
|e e |
-1 + a*|----- + ----| / 8*I*a -8*I*a\
\ 2 2 / I*\- e + e /*cot(8*a)
- ------------------------- + -------------------------------
2 -8*I*a 8*I*a
/ -6*I*a 6*I*a\ e + e
\- e + e /
1 + ---------------------
4
$$- \frac{a \left(\frac{e^{6 i}}{2} + \frac{e^{- 6 i}}{2}\right)^{2} - 1}{\frac{\left(e^{6 i a} - e^{- 6 i a}\right)^{2}}{4} + 1} + \frac{i \left(- e^{8 i a} + e^{- 8 i a}\right) \cot{\left(8 a \right)}}{e^{8 i a} + e^{- 8 i a}}$$
-(-1 + a*(exp(-6*i)/2 + exp(6*i)/2)^2)/(1 + (-exp(-6*i*a) + exp(6*i*a))^2/4) + i*(-exp(8*i*a) + exp(-8*i*a))*cot(8*a)/(exp(-8*i*a) + exp(8*i*a))
Combinatoria
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2 2
-1 + a*cos (6) - cot(8*a)*tan(8*a) + sin (6*a)*cot(8*a)*tan(8*a)
----------------------------------------------------------------
(1 + sin(6*a))*(-1 + sin(6*a))
$$\frac{a \cos^{2}{\left(6 \right)} + \sin^{2}{\left(6 a \right)} \tan{\left(8 a \right)} \cot{\left(8 a \right)} - \tan{\left(8 a \right)} \cot{\left(8 a \right)} - 1}{\left(\sin{\left(6 a \right)} - 1\right) \left(\sin{\left(6 a \right)} + 1\right)}$$
(-1 + a*cos(6)^2 - cot(8*a)*tan(8*a) + sin(6*a)^2*cot(8*a)*tan(8*a))/((1 + sin(6*a))*(-1 + sin(6*a)))
Parte trigonométrica
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2
-1 + a*cos (6)
1 - --------------
2
1 - sin (6*a)
$$1 - \frac{a \cos^{2}{\left(6 \right)} - 1}{1 - \sin^{2}{\left(6 a \right)}}$$
/ / 1 \\
| a*|1 + -------||
| \ sec(12)/|
2*|-1 + ---------------|
\ 2 /
1 - ------------------------
1
1 + ---------
sec(12*a)
$$1 - \frac{2 \left(\frac{a \left(\frac{1}{\sec{\left(12 \right)}} + 1\right)}{2} - 1\right)}{1 + \frac{1}{\sec{\left(12 a \right)}}}$$
/ / / pi\\\
| a*|1 + sin|12 + --|||
| \ \ 2 //|
2*|-1 + --------------------|
\ 2 /
1 - -----------------------------
/pi \
1 + sin|-- + 12*a|
\2 /
$$- \frac{2 \left(\frac{a \left(\sin{\left(\frac{\pi}{2} + 12 \right)} + 1\right)}{2} - 1\right)}{\sin{\left(12 a + \frac{\pi}{2} \right)} + 1} + 1$$
/ 2 \
2*\-1 + a*cos (6)/
1 - ------------------
1 + cos(12*a)
$$- \frac{2 \left(a \cos^{2}{\left(6 \right)} - 1\right)}{\cos{\left(12 a \right)} + 1} + 1$$
a
-1 + -------------
2/ pi\
csc |-6 + --|
\ 2 /
1 - ------------------
1
1 - ---------
2
csc (6*a)
$$1 - \frac{\frac{a}{\csc^{2}{\left(-6 + \frac{\pi}{2} \right)}} - 1}{1 - \frac{1}{\csc^{2}{\left(6 a \right)}}}$$
/ 2\
| / 2 \ |
| a*\-1 + cot (3)/ |
2*|-1 + -----------------|
| 2 |
| / 2 \ |
\ \1 + cot (3)/ /
1 - --------------------------
2
-1 + cot (6*a)
1 + --------------
2
1 + cot (6*a)
$$1 - \frac{2 \left(\frac{a \left(-1 + \cot^{2}{\left(3 \right)}\right)^{2}}{\left(1 + \cot^{2}{\left(3 \right)}\right)^{2}} - 1\right)}{\frac{\cot^{2}{\left(6 a \right)} - 1}{\cot^{2}{\left(6 a \right)} + 1} + 1}$$
/ / 2 \\
| | 1 - tan (6)||
| a*|1 + -----------||
| | 2 ||
| \ 1 + tan (6)/|
2*|-1 + -------------------|
\ 2 /
1 - ----------------------------
2
1 - tan (6*a)
1 + -------------
2
1 + tan (6*a)
$$- \frac{2 \left(\frac{a \left(\frac{1 - \tan^{2}{\left(6 \right)}}{\tan^{2}{\left(6 \right)} + 1} + 1\right)}{2} - 1\right)}{\frac{1 - \tan^{2}{\left(6 a \right)}}{\tan^{2}{\left(6 a \right)} + 1} + 1} + 1$$
2
/ 2 \
a*\1 - tan (3)/
-1 + ----------------
2
/ 2 \
\1 + tan (3)/
1 - ---------------------
2
4*tan (3*a)
1 - ----------------
2
/ 2 \
\1 + tan (3*a)/
$$1 - \frac{\frac{a \left(1 - \tan^{2}{\left(3 \right)}\right)^{2}}{\left(\tan^{2}{\left(3 \right)} + 1\right)^{2}} - 1}{1 - \frac{4 \tan^{2}{\left(3 a \right)}}{\left(\tan^{2}{\left(3 a \right)} + 1\right)^{2}}}$$
2
-1 + a*cos (6)
1 - ------------------
2/ pi\
1 - cos |6*a - --|
\ 2 /
$$1 - \frac{a \cos^{2}{\left(6 \right)} - 1}{1 - \cos^{2}{\left(6 a - \frac{\pi}{2} \right)}}$$
/ / 1 \\
| a*|1 + -------------||
| | / pi\||
| | csc|-12 + --|||
| \ \ 2 //|
2*|-1 + ---------------------|
\ 2 /
1 - ------------------------------
1
1 + --------------
/pi \
csc|-- - 12*a|
\2 /
$$1 - \frac{2 \left(\frac{a \left(\frac{1}{\csc{\left(-12 + \frac{\pi}{2} \right)}} + 1\right)}{2} - 1\right)}{1 + \frac{1}{\csc{\left(- 12 a + \frac{\pi}{2} \right)}}}$$
a
-1 + -------
2
sec (6)
1 - ------------------
1
1 - --------------
2/ pi\
sec |6*a - --|
\ 2 /
$$1 - \frac{\frac{a}{\sec^{2}{\left(6 \right)}} - 1}{1 - \frac{1}{\sec^{2}{\left(6 a - \frac{\pi}{2} \right)}}}$$
/ 2/ pi\\
2*|-1 + a*sin |6 + --||
\ \ 2 //
1 - -----------------------
/pi \
1 + sin|-- + 12*a|
\2 /
$$- \frac{2 \left(a \sin^{2}{\left(\frac{\pi}{2} + 6 \right)} - 1\right)}{\sin{\left(12 a + \frac{\pi}{2} \right)} + 1} + 1$$
a
-1 + -------
2
sec (6)
1 - -------------
1
1 - ---------
2
csc (6*a)
$$1 - \frac{\frac{a}{\sec^{2}{\left(6 \right)}} - 1}{1 - \frac{1}{\csc^{2}{\left(6 a \right)}}}$$
/ a*(1 + cos(12))\
2*|-1 + ---------------|
\ 2 /
1 - ------------------------
1 + cos(12*a)
$$- \frac{2 \left(\frac{a \left(\cos{\left(12 \right)} + 1\right)}{2} - 1\right)}{\cos{\left(12 a \right)} + 1} + 1$$
/ a \
2*|-1 + -------|
| 2 |
\ sec (6)/
1 - ----------------
1
1 + ---------
sec(12*a)
$$1 - \frac{2 \left(\frac{a}{\sec^{2}{\left(6 \right)}} - 1\right)}{1 + \frac{1}{\sec{\left(12 a \right)}}}$$
2
/ 2 \
a*\-1 + cot (3)/
-1 + -----------------
2
/ 2 \
\1 + cot (3)/
1 - ----------------------
2
4*cot (3*a)
1 - ----------------
2
/ 2 \
\1 + cot (3*a)/
$$1 - \frac{\frac{a \left(-1 + \cot^{2}{\left(3 \right)}\right)^{2}}{\left(1 + \cot^{2}{\left(3 \right)}\right)^{2}} - 1}{1 - \frac{4 \cot^{2}{\left(3 a \right)}}{\left(\cot^{2}{\left(3 a \right)} + 1\right)^{2}}}$$
/ 2\
| / 2 \ |
| a*\1 - tan (3)/ |
2*|-1 + ----------------|
| 2 |
| / 2 \ |
\ \1 + tan (3)/ /
1 - -------------------------
2
1 - tan (6*a)
1 + -------------
2
1 + tan (6*a)
$$1 - \frac{2 \left(\frac{a \left(1 - \tan^{2}{\left(3 \right)}\right)^{2}}{\left(\tan^{2}{\left(3 \right)} + 1\right)^{2}} - 1\right)}{\frac{1 - \tan^{2}{\left(6 a \right)}}{\tan^{2}{\left(6 a \right)} + 1} + 1}$$
/ a \
2*|-1 + -------------|
| 2/ pi\|
| csc |-6 + --||
\ \ 2 //
1 - ----------------------
1
1 + --------------
/pi \
csc|-- - 12*a|
\2 /
$$1 - \frac{2 \left(\frac{a}{\csc^{2}{\left(-6 + \frac{\pi}{2} \right)}} - 1\right)}{1 + \frac{1}{\csc{\left(- 12 a + \frac{\pi}{2} \right)}}}$$
2/ pi\
-1 + a*sin |6 + --|
\ 2 /
1 - -------------------
2
1 - sin (6*a)
$$1 - \frac{a \sin^{2}{\left(\frac{\pi}{2} + 6 \right)} - 1}{1 - \sin^{2}{\left(6 a \right)}}$$
/ / 2 \\
| | -1 + cot (6)||
| a*|1 + ------------||
| | 2 ||
| \ 1 + cot (6) /|
2*|-1 + --------------------|
\ 2 /
1 - -----------------------------
2
-1 + cot (6*a)
1 + --------------
2
1 + cot (6*a)
$$- \frac{2 \left(\frac{a \left(\frac{-1 + \cot^{2}{\left(6 \right)}}{1 + \cot^{2}{\left(6 \right)}} + 1\right)}{2} - 1\right)}{\frac{\cot^{2}{\left(6 a \right)} - 1}{\cot^{2}{\left(6 a \right)} + 1} + 1} + 1$$
1 - 2*(-1 + a*(1 + (-1 + cot(6)^2)/(1 + cot(6)^2))/2)/(1 + (-1 + cot(6*a)^2)/(1 + cot(6*a)^2))