Sr Examen
¿Cómo vas a descomponer esta tgA*ctgA-cos^2A/(2sinA) expresión en fracciones?
Solución
Ha introducido
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2
cos (a)
tan(a)*cot(a) - --------
2*sin(a)
$$\tan{\left(a \right)} \cot{\left(a \right)} - \frac{\cos^{2}{\left(a \right)}}{2 \sin{\left(a \right)}}$$
tan(a)*cot(a) - cos(a)^2/(2*sin(a))
Simplificación general
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sin(a) 1
1 + ------ - --------
2 2*sin(a)
$$\frac{\sin{\left(a \right)}}{2} + 1 - \frac{1}{2 \sin{\left(a \right)}}$$
1 + sin(a)/2 - 1/(2*sin(a))
Unión de expresiones racionales
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2
- cos (a) + 2*cot(a)*sin(a)*tan(a)
----------------------------------
2*sin(a)
$$\frac{2 \sin{\left(a \right)} \tan{\left(a \right)} \cot{\left(a \right)} - \cos^{2}{\left(a \right)}}{2 \sin{\left(a \right)}}$$
(-cos(a)^2 + 2*cot(a)*sin(a)*tan(a))/(2*sin(a))
Denominador racional
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2
- cos (a) + 2*cot(a)*sin(a)*tan(a)
----------------------------------
2*sin(a)
$$\frac{2 \sin{\left(a \right)} \tan{\left(a \right)} \cot{\left(a \right)} - \cos^{2}{\left(a \right)}}{2 \sin{\left(a \right)}}$$
(-cos(a)^2 + 2*cot(a)*sin(a)*tan(a))/(2*sin(a))
Combinatoria
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/ 2 \
-\cos (a) - 2*cot(a)*sin(a)*tan(a)/
------------------------------------
2*sin(a)
$$- \frac{- 2 \sin{\left(a \right)} \tan{\left(a \right)} \cot{\left(a \right)} + \cos^{2}{\left(a \right)}}{2 \sin{\left(a \right)}}$$
-(cos(a)^2 - 2*cot(a)*sin(a)*tan(a))/(2*sin(a))
Potencias
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2
/ I*a -I*a\
|e e |
I*|---- + -----| / I*a -I*a\
\ 2 2 / I*\- e + e /*cot(a)
- ----------------- + -------------------------
-I*a I*a I*a -I*a
- e + e e + e
$$\frac{i \left(- e^{i a} + e^{- i a}\right) \cot{\left(a \right)}}{e^{i a} + e^{- i a}} - \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}}{e^{i a} - e^{- i a}}$$
-i*(exp(i*a)/2 + exp(-i*a)/2)^2/(-exp(-i*a) + exp(i*a)) + i*(-exp(i*a) + exp(-i*a))*cot(a)/(exp(i*a) + exp(-i*a))
Parte trigonométrica
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/ pi\
sec|a - --|
1 \ 2 /
1 + ------------- - -----------
/ pi\ 2
2*sec|a - --|
\ 2 /
$$- \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{2} + 1 + \frac{1}{2 \sec{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\
sec|a - --|
\ 2 /
1 - -----------
2
2*sec (a)
$$1 - \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{2 \sec^{2}{\left(a \right)}}$$
2
/ 2/a\\
|1 - tan |-||
\ \2//
1 - ----------------------
/ 2/a\\ /a\
4*|1 + tan |-||*tan|-|
\ \2// \2/
$$- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{4 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}} + 1$$
/ pi\
cos|a - --|
\ 2 / 1
1 + ----------- - -------------
2 / pi\
2*cos|a - --|
\ 2 /
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{2} + 1 - \frac{1}{2 \cos{\left(a - \frac{\pi}{2} \right)}}$$
sin(a) 1
1 + ------ - --------
2 2*sin(a)
$$\frac{\sin{\left(a \right)}}{2} + 1 - \frac{1}{2 \sin{\left(a \right)}}$$
/pi \
sec|-- - a|
\2 /
1 - -----------
2
2*sec (a)
$$1 - \frac{\sec{\left(- a + \frac{\pi}{2} \right)}}{2 \sec^{2}{\left(a \right)}}$$
2
cos (a)
1 - --------
2*sin(a)
$$1 - \frac{\cos^{2}{\left(a \right)}}{2 \sin{\left(a \right)}}$$
2
cos (a)
1 - -------------
/ pi\
2*cos|a - --|
\ 2 /
$$- \frac{\cos^{2}{\left(a \right)}}{2 \cos{\left(a - \frac{\pi}{2} \right)}} + 1$$
csc(a)
1 - ---------
2
2*sec (a)
$$- \frac{\csc{\left(a \right)}}{2 \sec^{2}{\left(a \right)}} + 1$$
/a\ 2/a\
tan|-| 1 + tan |-|
\2/ \2/
1 + ----------- - -----------
2/a\ /a\
1 + tan |-| 4*tan|-|
\2/ \2/
$$- \frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{4 \tan{\left(\frac{a}{2} \right)}} + 1 + \frac{\tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
sin(a) csc(a)
1 + ------ - ------
2 2
$$\frac{\sin{\left(a \right)}}{2} - \frac{\csc{\left(a \right)}}{2} + 1$$
1 csc(a)
1 + -------- - ------
2*csc(a) 2
$$- \frac{\csc{\left(a \right)}}{2} + 1 + \frac{1}{2 \csc{\left(a \right)}}$$
csc(a)
1 - --------------
2/pi \
2*csc |-- - a|
\2 /
$$- \frac{\csc{\left(a \right)}}{2 \csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1$$
/a\ 2/a\
cot|-| 1 + cot |-|
\2/ \2/
1 + ----------- - -----------
2/a\ /a\
1 + cot |-| 4*cot|-|
\2/ \2/
$$- \frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{4 \cot{\left(\frac{a}{2} \right)}} + 1 + \frac{\cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}$$
2/ pi\
sin |a + --|
\ 2 /
1 - ------------
2*sin(a)
$$1 - \frac{\sin^{2}{\left(a + \frac{\pi}{2} \right)}}{2 \sin{\left(a \right)}}$$
2
/ 2/a\\
|-1 + cot |-||
\ \2//
1 - ----------------------
/ 2/a\\ /a\
4*|1 + cot |-||*cot|-|
\ \2// \2/
$$- \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{4 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}} + 1$$
2
cos (a)*csc(a)
1 - --------------
2
$$- \frac{\cos^{2}{\left(a \right)} \csc{\left(a \right)}}{2} + 1$$
1 - cos(a)^2*csc(a)/2