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