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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(b)/((cos*b/2)-(sin*b/2))

¿Cómo vas a descomponer esta cos(b)/((cos*b/2)-(sin*b/2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
     cos(b)    
---------------
cos(b)   sin(b)
------ - ------
  2        2
cos⁡(b)−sin⁡(b)2+cos⁡(b)2\frac{\cos{\left(b \right)}}{- \frac{\sin{\left(b \right)}}{2} + \frac{\cos{\left(b \right)}}{2}} 
cos(b)/(cos(b)/2 - sin(b)/2)
Simplificación general [src] 
  ___       
\/ 2 *cos(b)
------------
   /    pi\ 
cos|b + --| 
   \    4 /
2cos⁡(b)cos⁡(b+π4)\frac{\sqrt{2} \cos{\left(b \right)}}{\cos{\left(b + \frac{\pi}{4} \right)}} 
sqrt(2)*cos(b)/cos(b + pi/4)
Denominador común [src] 
        2*sin(b)    
2 + ----------------
    -sin(b) + cos(b)
2+2sin⁡(b)−sin⁡(b)+cos⁡(b)2 + \frac{2 \sin{\left(b \right)}}{- \sin{\left(b \right)} + \cos{\left(b \right)}} 
2 + 2*sin(b)/(-sin(b) + cos(b))
Respuesta numérica [src] 
cos(b)/(0.5*cos(b) - 0.5*sin(b))
cos(b)/(0.5*cos(b) - 0.5*sin(b))
Potencias [src] 
            I*b    -I*b          
           e      e              
           ---- + -----          
            2       2            
---------------------------------
 I*b    -I*b     /   -I*b    I*b\
e      e       I*\- e     + e   /
---- + ----- + ------------------
 4       4             4
eib2+e−ib2i(eib−e−ib)4+eib4+e−ib4\frac{\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}}{\frac{i \left(e^{i b} - e^{- i b}\right)}{4} + \frac{e^{i b}}{4} + \frac{e^{- i b}}{4}} 
(exp(i*b)/2 + exp(-i*b)/2)/(exp(i*b)/4 + exp(-i*b)/4 + i*(-exp(-i*b) + exp(i*b))/4)
Denominador racional [src] 
    2*cos(b)    
----------------
-sin(b) + cos(b)
2cos⁡(b)−sin⁡(b)+cos⁡(b)\frac{2 \cos{\left(b \right)}}{- \sin{\left(b \right)} + \cos{\left(b \right)}} 
2*cos(b)/(-sin(b) + cos(b))
Combinatoria [src] 
    2*cos(b)    
----------------
-sin(b) + cos(b)
2cos⁡(b)−sin⁡(b)+cos⁡(b)\frac{2 \cos{\left(b \right)}}{- \sin{\left(b \right)} + \cos{\left(b \right)}} 
2*cos(b)/(-sin(b) + cos(b))
Unión de expresiones racionales [src] 
    2*cos(b)    
----------------
-sin(b) + cos(b)
2cos⁡(b)−sin⁡(b)+cos⁡(b)\frac{2 \cos{\left(b \right)}}{- \sin{\left(b \right)} + \cos{\left(b \right)}} 
2*cos(b)/(-sin(b) + cos(b))
Parte trigonométrica [src] 
  ___ /       2/b   pi\\ /        2/b\\
\/ 2 *|1 + cot |- + --||*|-1 + cot |-||
      \        \2   8 // \         \2//
---------------------------------------
   /       2/b\\ /        2/b   pi\\   
   |1 + cot |-||*|-1 + cot |- + --||   
   \        \2// \         \2   8 //
2(cot⁡2(b2)−1)(cot⁡2(b2+π8)+1)(cot⁡2(b2)+1)(cot⁡2(b2+π8)−1)\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{8} \right)} + 1\right)}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{8} \right)} - 1\right)} 
  ___    /    pi\
\/ 2 *sec|b + --|
         \    4 /
-----------------
      sec(b)
2sec⁡(b+π4)sec⁡(b)\frac{\sqrt{2} \sec{\left(b + \frac{\pi}{4} \right)}}{\sec{\left(b \right)}} 
  ___    /     pi\
\/ 2 *csc|-b + --|
         \     4 /
------------------
      /pi    \    
   csc|-- - b|    
      \2     /
2csc⁡(−b+π4)csc⁡(−b+π2)\frac{\sqrt{2} \csc{\left(- b + \frac{\pi}{4} \right)}}{\csc{\left(- b + \frac{\pi}{2} \right)}} 
                        2/b\                 
                 1 - tan |-|                 
                         \2/                 
---------------------------------------------
              /         2/b\           /b\  \
              |  1 - tan |-|        tan|-|  |
/       2/b\\ |          \2/           \2/  |
|1 + tan |-||*|--------------- - -----------|
\        \2// |  /       2/b\\          2/b\|
              |2*|1 + tan |-||   1 + tan |-||
              \  \        \2//           \2//
1−tan⁡2(b2)(1−tan⁡2(b2)2(tan⁡2(b2)+1)−tan⁡(b2)tan⁡2(b2)+1)(tan⁡2(b2)+1)\frac{1 - \tan^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{b}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} - \frac{\tan{\left(\frac{b}{2} \right)}}{\tan^{2}{\left(\frac{b}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} 
  ___    /    pi\
\/ 2 *sin|b + --|
         \    2 /
-----------------
     /    3*pi\  
  sin|b + ----|  
     \     4  /
2sin⁡(b+π2)sin⁡(b+3π4)\frac{\sqrt{2} \sin{\left(b + \frac{\pi}{2} \right)}}{\sin{\left(b + \frac{3 \pi}{4} \right)}} 
       cos(b)       
--------------------
            /    pi\
         cos|b - --|
cos(b)      \    2 /
------ - -----------
  2           2
cos⁡(b)cos⁡(b)2−cos⁡(b−π2)2\frac{\cos{\left(b \right)}}{\frac{\cos{\left(b \right)}}{2} - \frac{\cos{\left(b - \frac{\pi}{2} \right)}}{2}} 
  ___       
\/ 2 *cos(b)
------------
   /    pi\ 
cos|b + --| 
   \    4 /
2cos⁡(b)cos⁡(b+π4)\frac{\sqrt{2} \cos{\left(b \right)}}{\cos{\left(b + \frac{\pi}{4} \right)}} 
                  1                   
--------------------------------------
/      1            1    \    /pi    \
|------------- - --------|*csc|-- - b|
|     /pi    \   2*csc(b)|    \2     /
|2*csc|-- - b|           |            
\     \2     /           /
1(12csc⁡(−b+π2)−12csc⁡(b))csc⁡(−b+π2)\frac{1}{\left(\frac{1}{2 \csc{\left(- b + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(b \right)}}\right) \csc{\left(- b + \frac{\pi}{2} \right)}} 
                         2/b\                
                 -1 + cot |-|                
                          \2/                
---------------------------------------------
              /          2/b\          /b\  \
              |  -1 + cot |-|       cot|-|  |
/       2/b\\ |           \2/          \2/  |
|1 + cot |-||*|--------------- - -----------|
\        \2// |  /       2/b\\          2/b\|
              |2*|1 + cot |-||   1 + cot |-||
              \  \        \2//           \2//
cot⁡2(b2)−1(cot⁡2(b2)−12(cot⁡2(b2)+1)−cot⁡(b2)cot⁡2(b2)+1)(cot⁡2(b2)+1)\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{2 \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)} - \frac{\cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)} 
  ___           /    pi\
\/ 2 *cos(b)*sec|b + --|
                \    4 /
2cos⁡(b)sec⁡(b+π4)\sqrt{2} \cos{\left(b \right)} \sec{\left(b + \frac{\pi}{4} \right)} 
  ___           /     pi\
\/ 2 *cos(b)*csc|-b + --|
                \     4 /
2cos⁡(b)csc⁡(−b+π4)\sqrt{2} \cos{\left(b \right)} \csc{\left(- b + \frac{\pi}{4} \right)} 
             1              
----------------------------
/   1          1    \       
|-------- - --------|*sec(b)
\2*sec(b)   2*csc(b)/
1(12sec⁡(b)−12csc⁡(b))sec⁡(b)\frac{1}{\left(\frac{1}{2 \sec{\left(b \right)}} - \frac{1}{2 \csc{\left(b \right)}}\right) \sec{\left(b \right)}} 
       /    pi\     
    sin|b + --|     
       \    2 /     
--------------------
   /    pi\         
sin|b + --|         
   \    2 /   sin(b)
----------- - ------
     2          2
sin⁡(b+π2)−sin⁡(b)2+sin⁡(b+π2)2\frac{\sin{\left(b + \frac{\pi}{2} \right)}}{- \frac{\sin{\left(b \right)}}{2} + \frac{\sin{\left(b + \frac{\pi}{2} \right)}}{2}} 
                1                
---------------------------------
/   1             1      \       
|-------- - -------------|*sec(b)
|2*sec(b)        /    pi\|       
|           2*sec|b - --||       
\                \    2 //
1(−12sec⁡(b−π2)+12sec⁡(b))sec⁡(b)\frac{1}{\left(- \frac{1}{2 \sec{\left(b - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(b \right)}}\right) \sec{\left(b \right)}} 
  ___ /       2/b   pi\\ /       2/b\\
\/ 2 *|1 + tan |- + --||*|1 - tan |-||
      \        \2   8 // \        \2//
--------------------------------------
   /       2/b\\ /       2/b   pi\\   
   |1 + tan |-||*|1 - tan |- + --||   
   \        \2// \        \2   8 //
2(1−tan⁡2(b2))(tan⁡2(b2+π8)+1)(1−tan⁡2(b2+π8))(tan⁡2(b2)+1)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{b}{2} \right)}\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{8} \right)} + 1\right)}{\left(1 - \tan^{2}{\left(\frac{b}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} 
                        ___                          
                   -2*\/ 2 *cos(b)                   
-----------------------------------------------------
                    ___________    ___________       
    ___            /       ___    /       ___        
- \/ 2 *cos(b) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(b)
−22cos⁡(b)2−22+2sin⁡(b)−2cos⁡(b)- \frac{2 \sqrt{2} \cos{\left(b \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(b \right)} - \sqrt{2} \cos{\left(b \right)}} 
  ___    2/b\ /       2/b   pi\\ /       2/b\\
\/ 2 *cos |-|*|1 + tan |- + --||*|1 - tan |-||
          \2/ \        \2   8 // \        \2//
----------------------------------------------
                      2/b   pi\               
               1 - tan |- + --|               
                       \2   8 /
2(1−tan⁡2(b2))(tan⁡2(b2+π8)+1)cos⁡2(b2)1−tan⁡2(b2+π8)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{b}{2} \right)}\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(\frac{b}{2} \right)}}{1 - \tan^{2}{\left(\frac{b}{2} + \frac{\pi}{8} \right)}} 
sqrt(2)*cos(b/2)^2*(1 + tan(b/2 + pi/8)^2)*(1 - tan(b/2)^2)/(1 - tan(b/2 + pi/8)^2)
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