Sr Examen
¿Cómo vas a descomponer esta cos(a+b)/((cos(a)*sin(b))) expresión en fracciones?
Solución
Ha introducido
[src]
cos(a + b) ------------- cos(a)*sin(b)
$$\frac{\cos{\left(a + b \right)}}{\sin{\left(b \right)} \cos{\left(a \right)}}$$
cos(a + b)/((cos(a)*sin(b)))
Simplificación general
[src]
1 ------ - tan(a) tan(b)
$$- \tan{\left(a \right)} + \frac{1}{\tan{\left(b \right)}}$$
1/tan(b) - tan(a)
Potencias
[src]
/ I*(a + b) I*(-a - b)\
|e e |
2*I*|---------- + -----------|
\ 2 2 /
-------------------------------
/ I*a -I*a\
|e e | / -I*b I*b\
|---- + -----|*\- e + e /
\ 2 2 /
$$\frac{2 i \left(\frac{e^{i \left(- a - b\right)}}{2} + \frac{e^{i \left(a + b\right)}}{2}\right)}{\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i b} - e^{- i b}\right)}$$
2*i*(exp(i*(a + b))/2 + exp(i*(-a - b))/2)/((exp(i*a)/2 + exp(-i*a)/2)*(-exp(-i*b) + exp(i*b)))
Parte trigonométrica
[src]
2
sin(2*b) 2*sin (a)
--------- - ---------
2 sin(2*a)
2*sin (b)
$$- \frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(2 b \right)}}{2 \sin^{2}{\left(b \right)}}$$
2/a b\
1 - tan |- + -|
\2 2/
--------------------------------------------------------------------------------------
2 2
/ 2/a b\\ / 2/a\\ / 2/a\\ / 2/b\\ 4/a\ 4/b\ /b\
2*|1 + tan |- + -||*|1 - tan |-||*|1 - tan |-|| *|1 - tan |-|| *cos |-|*cos |-|*tan|-|
\ \2 2// \ \2// \ \4// \ \4// \4/ \4/ \2/
$$\frac{1 - \tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \left(1 - \tan^{2}{\left(\frac{b}{4} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right) \cos^{4}{\left(\frac{a}{4} \right)} \cos^{4}{\left(\frac{b}{4} \right)} \tan{\left(\frac{b}{2} \right)}}$$
/ pi\
sec|b - --|
\ 2 / sec(a)
----------- - -----------
sec(b) / pi\
sec|a - --|
\ 2 /
$$- \frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + \frac{\sec{\left(b - \frac{\pi}{2} \right)}}{\sec{\left(b \right)}}$$
2 2
/ 2/a\\ / 2/b\\ 2/a b\ / 2/a b\\
|1 + cot |-|| *|1 + cot |-|| *cot |- + -|*|-1 + cot |- + -||
\ \4// \ \4// \4 4/ \ \2 2//
------------------------------------------------------------
2
/ 2/a b\\ / 2/a\\ /b\ 2/a\ 2/b\
8*|1 + cot |- + -|| *|-1 + cot |-||*cot|-|*cot |-|*cot |-|
\ \4 4// \ \2// \2/ \4/ \4/
$$\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} - 1\right) \cot^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)}}{8 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)} + 1\right)^{2} \cot^{2}{\left(\frac{a}{4} \right)} \cot^{2}{\left(\frac{b}{4} \right)} \cot{\left(\frac{b}{2} \right)}}$$
2/a b\ 4/a + b\ 2/a\ 2/b\ / 2/a b\\ /b\
cot |- + -|*sin |-----|*tan |-|*tan |-|*|-1 + cot |- + -||*tan|-|
\4 4/ \ 4 / \4/ \4/ \ \2 2// \2/
-----------------------------------------------------------------
/ 2/a\\ 4/a\ 4/b\
8*|-1 + cot |-||*sin |-|*sin |-|
\ \2// \4/ \4/
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} - 1\right) \sin^{4}{\left(\frac{a + b}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{b}{4} \right)} \tan{\left(\frac{b}{2} \right)} \cot^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)}}{8 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin^{4}{\left(\frac{a}{4} \right)} \sin^{4}{\left(\frac{b}{4} \right)}}$$
/ 2/a b\ \
| csc |- + -| |
2/a\ | \2 2/ | /b\ /pi b\
csc |-|*|-1 + ----------------|*csc|-|*csc|-- - -|
\2/ | 2/pi a b\| \2/ \2 2/
| csc |-- - - - -||
\ \2 2 2//
--------------------------------------------------
/ 2/a\ \
| csc |-| |
| \2/ | 2/a b\
2*|-1 + ------------|*csc |- + -|
| 2/pi a\| \2 2/
| csc |-- - -||
\ \2 2//
$$\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc^{2}{\left(\frac{a}{2} \right)} \csc{\left(\frac{b}{2} \right)} \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{2 \left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}$$
/pi \
csc|-- - a|
csc(b) \2 /
----------- - -----------
/pi \ csc(a)
csc|-- - b|
\2 /
$$\frac{\csc{\left(b \right)}}{\csc{\left(- b + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$
/ 1 \
-|1 - ------|*cos(a + b)
\ cos(a)/
-------------------------
/b\ 2/a\ /b\
4*cos|-|*sin |-|*sin|-|
\2/ \2/ \2/
$$- \frac{\left(1 - \frac{1}{\cos{\left(a \right)}}\right) \cos{\left(a + b \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(\frac{b}{2} \right)} \cos{\left(\frac{b}{2} \right)}}$$
2/a b pi\
cos |- + - - --|
\2 2 2 /
1 - ----------------
2/a b\
cos |- + -|
\2 2/
----------------------------------------------------------------------
/ 2/a b pi\\ / 2/a pi\\
| cos |- + - - --|| | cos |- - --||
| \2 2 2 /| | \2 2 /| 2/a\ /b\ /b pi\
2*|1 + ----------------|*|1 - ------------|*cos |-|*cos|-|*cos|- - --|
| 2/a b\ | | 2/a\ | \2/ \2/ \2 2 /
| cos |- + -| | | cos |-| |
\ \2 2/ / \ \2/ /
$$\frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}}{2 \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cos^{2}{\left(\frac{a}{2} \right)} \cos{\left(\frac{b}{2} \right)} \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}$$
2 2
/ 2/a\\ / 2/b\\ / 2/a b\\
|1 + tan |-|| *|1 + tan |-|| *|1 - tan |- + -||
\ \4// \ \4// \ \2 2//
----------------------------------------------------------------------
2 2
/ 2/a b\\ / 2/a\\ / 2/a\\ / 2/b\\ /b\
2*|1 + tan |- + -||*|1 - tan |-||*|1 - tan |-|| *|1 - tan |-|| *tan|-|
\ \2 2// \ \2// \ \4// \ \4// \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{b}{4} \right)} + 1\right)^{2}}{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \left(1 - \tan^{2}{\left(\frac{b}{4} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right) \tan{\left(\frac{b}{2} \right)}}$$
/pi \
csc(b)*csc|-- - a|
\2 /
------------------
/pi \
csc|-- - a - b|
\2 /
$$\frac{\csc{\left(b \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(- a - b + \frac{\pi}{2} \right)}}$$
2 2
/ 2/a\\ / 2/b\\ / 1 \ /b\
|1 + cot |-|| *|1 + cot |-|| *|1 - -----------|*cot|-|
\ \4// \ \4// | 2/a b\| \2/
| cot |- + -||
\ \2 2//
-----------------------------------------------------------------
2 2
/ 1 \ / 1 \ / 2/a\\ / 2/b\\
2*|1 + -----------|*|1 - -------|*|-1 + cot |-|| *|-1 + cot |-||
| 2/a b\| | 2/a\| \ \4// \ \4//
| cot |- + -|| | cot |-||
\ \2 2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{4} \right)} + 1\right)^{2} \cot{\left(\frac{b}{2} \right)}}{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{4} \right)} - 1\right)^{2}}$$
2*cos(a + b) ------------------------ -sin(a - b) + sin(a + b)
$$\frac{2 \cos{\left(a + b \right)}}{- \sin{\left(a - b \right)} + \sin{\left(a + b \right)}}$$
/ 2/a b\ \
| cos |- + -| |
2/a b pi\ | \2 2/ |
cos |- + - - --|*|-1 + ----------------|
\2 2 2 / | 2/a b pi\|
| cos |- + - - --||
\ \2 2 2 //
-----------------------------------------------------
/ 2/a\ \
| cos |-| |
| \2/ | /b\ 2/a pi\ /b pi\
2*|-1 + ------------|*cos|-|*cos |- - --|*cos|- - --|
| 2/a pi\| \2/ \2 2 / \2 2 /
| cos |- - --||
\ \2 2 //
$$\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{2 \left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(\frac{b}{2} \right)} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}$$
/ 4/a b\\
| 4*sin |- + -||
| \2 2/|
|1 - -------------|*sin(b)
| 2 |
\ sin (a + b) /
-----------------------------------------------------------------------
/ 4/a\\ / 4/a b\\
| 4*sin |-|| | 4*sin |- + -||
| \2/| | \2 2/| 2/b\ 2/pi a\ 2/pi b\
4*|1 - ---------|*|1 + -------------|*sin |-|*sin |-- + -|*sin |-- + -|
| 2 | | 2 | \2/ \2 2/ \2 2/
\ sin (a) / \ sin (a + b) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(b \right)}}{4 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin^{2}{\left(\frac{b}{2} \right)} \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} \sin^{2}{\left(\frac{b}{2} + \frac{\pi}{2} \right)}}$$
-4*(1 - cos(a + b))*cos(a + b) ------------------------------------------------------------------- -sin(2*a) - 2*sin(a + b) + 2*sin(a - b) + sin(2*b) + sin(2*(a + b))
$$- \frac{4 \left(1 - \cos{\left(a + b \right)}\right) \cos{\left(a + b \right)}}{- \sin{\left(2 a \right)} + \sin{\left(2 b \right)} + 2 \sin{\left(a - b \right)} - 2 \sin{\left(a + b \right)} + \sin{\left(2 \left(a + b\right) \right)}}$$
/ pi\
cos|a - --|
cos(b) \ 2 /
----------- - -----------
/ pi\ cos(a)
cos|b - --|
\ 2 /
$$\frac{\cos{\left(b \right)}}{\cos{\left(b - \frac{\pi}{2} \right)}} - \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$
/ pi\
sin|a + b + --|
\ 2 /
------------------
/ pi\
sin(b)*sin|a + --|
\ 2 /
$$\frac{\sin{\left(a + b + \frac{\pi}{2} \right)}}{\sin{\left(b \right)} \sin{\left(a + \frac{\pi}{2} \right)}}$$
/ 2/a b\\ /b\
|1 - tan |- + -||*cot|-|
\ \2 2// \2/
---------------------------------------------------------------------------------
2 2
/ 2/a b\\ / 2/a\\ / 2/a\\ / 2/b\\ 4/a\ 4/b\
2*|1 + tan |- + -||*|1 - tan |-||*|-1 + cot |-|| *|-1 + cot |-|| *sin |-|*sin |-|
\ \2 2// \ \2// \ \4// \ \4// \4/ \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}\right) \cot{\left(\frac{b}{2} \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{a}{4} \right)} \sin^{4}{\left(\frac{b}{4} \right)}}$$
/ 2/a b\ \
| sec |- + -| |
2/a\ | \2 2/ | /b\ /b pi\
sec |-|*|1 - ----------------|*sec|-|*sec|- - --|
\2/ | 2/a b pi\| \2/ \2 2 /
| sec |- + - - --||
\ \2 2 2 //
-------------------------------------------------
/ 2/a b\ \ / 2/a\ \
| sec |- + -| | | sec |-| |
| \2 2/ | | \2/ |
2*|1 + ----------------|*|1 - ------------|
| 2/a b pi\| | 2/a pi\|
| sec |- + - - --|| | sec |- - --||
\ \2 2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(\frac{a}{2} \right)} \sec{\left(\frac{b}{2} \right)} \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{2 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)}$$
-tan(a) + cot(b)
$$- \tan{\left(a \right)} + \cot{\left(b \right)}$$
cos(a + b)
------------------
/ pi\
cos(a)*cos|b - --|
\ 2 /
$$\frac{\cos{\left(a + b \right)}}{\cos{\left(a \right)} \cos{\left(b - \frac{\pi}{2} \right)}}$$
/ 2 \
2/a + b\ | sin (a + b) |
2*sin |-----|*|-1 + -------------|
\ 2 / | 4/a + b\|
| 4*sin |-----||
\ \ 2 //
------------------------------------
/ 2 \
| sin (a) |
(1 - cos(a))*|-1 + ---------|*sin(b)
| 4/a\|
| 4*sin |-||
\ \2//
$$\frac{2 \left(-1 + \frac{\sin^{2}{\left(a + b \right)}}{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}\right) \sin^{2}{\left(\frac{a + b}{2} \right)}}{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(1 - \cos{\left(a \right)}\right) \sin{\left(b \right)}}$$
/ pi\
sec(a)*sec|b - --|
\ 2 /
------------------
sec(a + b)
$$\frac{\sec{\left(a \right)} \sec{\left(b - \frac{\pi}{2} \right)}}{\sec{\left(a + b \right)}}$$
/ 2/pi a b\\
| csc |-- - - - -||
2/pi a\ | \2 2 2/| /b\ /pi b\
csc |-- - -|*|1 - ----------------|*csc|-|*csc|-- - -|
\2 2/ | 2/a b\ | \2/ \2 2/
| csc |- + -| |
\ \2 2/ /
------------------------------------------------------
/ 2/pi a b\\ / 2/pi a\\
| csc |-- - - - -|| | csc |-- - -||
| \2 2 2/| | \2 2/|
2*|1 + ----------------|*|1 - ------------|
| 2/a b\ | | 2/a\ |
| csc |- + -| | | csc |-| |
\ \2 2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \csc{\left(\frac{b}{2} \right)} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)} \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{2 \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right)}$$
2/a b\ / 1 \ / 1 \ /b\
tan |- + -|*|-1 + ------|*|-1 + -----------|*tan|-|
\4 4/ \ cos(a)/ | 2/a b\| \2/
| tan |- + -||
\ \2 2//
-----------------------------------------------------
2
/ 2/a b\\ 4/a\ 4/b\ 2/a\ 2/b\
16*|1 + tan |- + -|| *cos |-|*cos |-|*tan |-|*tan |-|
\ \4 4// \4/ \4/ \4/ \4/
$$\frac{\left(-1 + \frac{1}{\cos{\left(a \right)}}\right) \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \tan{\left(\frac{b}{2} \right)} \tan^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)}}{16 \left(\tan^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{a}{4} \right)} \cos^{4}{\left(\frac{b}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{b}{4} \right)}}$$
2/a b\
1 - tan |- + -|
\2 2/
--------------------------------------------------------
/ 2/a b\\ / 2/a\\ 2/a\ 2/b\ /b\
2*|1 + tan |- + -||*|1 - tan |-||*cos |-|*cos |-|*tan|-|
\ \2 2// \ \2// \2/ \2/ \2/
$$\frac{1 - \tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)} \cos^{2}{\left(\frac{b}{2} \right)} \tan{\left(\frac{b}{2} \right)}}$$
csc(b)*sec(a) ------------- sec(a + b)
$$\frac{\csc{\left(b \right)} \sec{\left(a \right)}}{\sec{\left(a + b \right)}}$$
2/a + b\ / 2/a b\\ /b\
sin |-----|*|-1 + cot |- + -||*tan|-|
\ 2 / \ \2 2// \2/
-------------------------------------
/ 2/a\\ 2/a\ 2/b\
2*|-1 + cot |-||*sin |-|*sin |-|
\ \2// \2/ \2/
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} - 1\right) \sin^{2}{\left(\frac{a + b}{2} \right)} \tan{\left(\frac{b}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{b}{2} \right)}}$$
1 ------ - tan(a) tan(b)
$$- \tan{\left(a \right)} + \frac{1}{\tan{\left(b \right)}}$$
/ 2/a\\ / 2/b\\ / 2/a b\\
|1 + cot |-||*|1 + cot |-||*|-1 + cot |- + -||
\ \2// \ \2// \ \2 2//
----------------------------------------------
/ 2/a b\\ / 2/a\\ /b\
2*|1 + cot |- + -||*|-1 + cot |-||*cot|-|
\ \2 2// \ \2// \2/
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right) \cot{\left(\frac{b}{2} \right)}}$$
/ 2 \
2/a b\ | sin (a + b) |
sin |- + -|*|-1 + -------------|
\2 2/ | 4/a b\|
| 4*sin |- + -||
\ \2 2//
--------------------------------
/ 2 \
| sin (a) | 2/a\
|-1 + ---------|*sin(b)*sin |-|
| 4/a\| \2/
| 4*sin |-||
\ \2//
$$\frac{\left(-1 + \frac{\sin^{2}{\left(a + b \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(b \right)}}$$
/ 4/a + b\\
| 4*sin |-----||
| \ 2 /|
2*|1 - -------------|*sin(b)
| 2 |
\ sin (a + b) /
--------------------------------------------------------------------------
/ 4/a\\ / 4/a + b\\
| 4*sin |-|| | 4*sin |-----||
| \2/| | \ 2 /|
(1 - cos(b))*|1 - ---------|*|1 + -------------|*(1 + cos(a))*(1 + cos(b))
| 2 | | 2 |
\ sin (a) / \ sin (a + b) /
$$\frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(b \right)}}{\left(1 - \cos{\left(b \right)}\right) \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \left(\cos{\left(a \right)} + 1\right) \left(\cos{\left(b \right)} + 1\right)}$$
2 2
/ 2/a\\ / 2/b\\ 2/a b\ / 1 \ /b\
|1 + tan |-|| *|1 + tan |-|| *tan |- + -|*|-1 + -----------|*tan|-|
\ \4// \ \4// \4 4/ | 2/a b\| \2/
| tan |- + -||
\ \2 2//
-------------------------------------------------------------------
2
/ 2/a b\\ / 1 \ 2/a\ 2/b\
8*|1 + tan |- + -|| *|-1 + -------|*tan |-|*tan |-|
\ \4 4// | 2/a\| \4/ \4/
| tan |-||
\ \2//
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{b}{4} \right)} + 1\right)^{2} \tan{\left(\frac{b}{2} \right)} \tan^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)}}{8 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{b}{4} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{b}{4} \right)}}$$
1 - ------ + cot(b) cot(a)
$$\cot{\left(b \right)} - \frac{1}{\cot{\left(a \right)}}$$
/ 2/a\\ / 2/b\\ / 2/a b\\
|1 + tan |-||*|1 + tan |-||*|1 - tan |- + -||
\ \2// \ \2// \ \2 2//
---------------------------------------------
/ 2/a b\\ / 2/a\\ /b\
2*|1 + tan |- + -||*|1 - tan |-||*tan|-|
\ \2 2// \ \2// \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right) \tan{\left(\frac{b}{2} \right)}}$$
/ 2/a b pi\\
| sec |- + - - --||
2/a pi\ | \2 2 2 /| /b\ /b pi\
sec |- - --|*|-1 + ----------------|*sec|-|*sec|- - --|
\2 2 / | 2/a b\ | \2/ \2 2 /
| sec |- + -| |
\ \2 2/ /
-------------------------------------------------------
/ 2/a pi\\
| sec |- - --||
| \2 2 /| 2/a b pi\
2*|-1 + ------------|*sec |- + - - --|
| 2/a\ | \2 2 2 /
| sec |-| |
\ \2/ /
$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \sec{\left(\frac{b}{2} \right)} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{2 \left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}$$
sec(a/2 - pi/2)^2*(-1 + sec(a/2 + b/2 - pi/2)^2/sec(a/2 + b/2)^2)*sec(b/2)*sec(b/2 - pi/2)/(2*(-1 + sec(a/2 - pi/2)^2/sec(a/2)^2)*sec(a/2 + b/2 - pi/2)^2)