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Least common denominator sin(pi/2-pi*a/(2*b))

An expression to simplify:

The solution

You have entered [src]
   /pi   pi*a\
sin|-- - ----|
   \2    2*b /
$$\sin{\left(\frac{\pi}{2} - \frac{\pi a}{2 b} \right)}$$
sin(pi/2 - pi*a/(2*b))
Fraction decomposition [src]
cos(pi*a/(2*b))
$$\cos{\left(\frac{\pi a}{2 b} \right)}$$
   /pi*a\
cos|----|
   \2*b /
General simplification [src]
   /pi*(b - a)\
sin|----------|
   \   2*b    /
$$\sin{\left(\frac{\pi \left(- a + b\right)}{2 b} \right)}$$
sin(pi*(b - a)/(2*b))
Combining rational expressions [src]
   /pi*(b - a)\
sin|----------|
   \   2*b    /
$$\sin{\left(\frac{\pi \left(- a + b\right)}{2 b} \right)}$$
sin(pi*(b - a)/(2*b))
Powers [src]
   /     /  pi   pi*a\      /pi   pi*a\\ 
   |   I*|- -- + ----|    I*|-- - ----|| 
   |     \  2    2*b /      \2    2*b /| 
-I*\- e                + e             / 
-----------------------------------------
                    2                    
$$- \frac{i \left(e^{i \left(- \frac{\pi a}{2 b} + \frac{\pi}{2}\right)} - e^{i \left(\frac{\pi a}{2 b} - \frac{\pi}{2}\right)}\right)}{2}$$
   /pi*a\
cos|----|
   \2*b /
$$\cos{\left(\frac{\pi a}{2 b} \right)}$$
cos(pi*a/(2*b))
Numerical answer [src]
sin(pi/2 - pi*a/(2*b))
sin(pi/2 - pi*a/(2*b))
Common denominator [src]
   /pi*a\
cos|----|
   \2*b /
$$\cos{\left(\frac{\pi a}{2 b} \right)}$$
cos(pi*a/(2*b))
Combinatorics [src]
   /pi*a\
cos|----|
   \2*b /
$$\cos{\left(\frac{\pi a}{2 b} \right)}$$
cos(pi*a/(2*b))
Rational denominator [src]
    /pi*(a - b)\
-sin|----------|
    \   2*b    /
$$- \sin{\left(\frac{\pi \left(a - b\right)}{2 b} \right)}$$
-sin(pi*(a - b)/(2*b))
Expand expression [src]
   /pi*a\
cos|----|
   \2*b /
$$\cos{\left(\frac{\pi a}{2 b} \right)}$$
    /  pi   pi*a\
-sin|- -- + ----|
    \  2    2*b /
$$- \sin{\left(\frac{\pi a}{2 b} - \frac{\pi}{2} \right)}$$
-sin(-pi/2 + pi*a/(2*b))
Trigonometric part [src]
        2/pi*a\
-1 + cot |----|
         \4*b /
---------------
        2/pi*a\
 1 + cot |----|
         \4*b /
$$\frac{\cot^{2}{\left(\frac{\pi a}{4 b} \right)} - 1}{\cot^{2}{\left(\frac{\pi a}{4 b} \right)} + 1}$$
                 2                  
------------------------------------
/           1       \    /pi   pi*a\
|1 + ---------------|*cot|-- + ----|
|       2/pi   pi*a\|    \4    4*b /
|    cot |-- + ----||               
\        \4    4*b //               
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}\right) \cot{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
               /pi   pi*a\          
          2*csc|-- - ----|          
               \4    4*b /          
------------------------------------
/       2/pi   pi*a\\               
|    csc |-- - ----||               
|        \4    4*b /|    /pi   pi*a\
|1 + ---------------|*csc|-- + ----|
|       2/pi   pi*a\|    \4    4*b /
|    csc |-- + ----||               
\        \4    4*b //               
$$\frac{2 \csc{\left(- \frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\left(\frac{\csc^{2}{\left(- \frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}} + 1\right) \csc{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
                 /pi   pi*a\            
            2*sec|-- + ----|            
                 \4    4*b /            
----------------------------------------
/        2/pi   pi*a\ \                 
|     sec |-- + ----| |                 
|         \4    4*b / |    /  pi   pi*a\
|1 + -----------------|*sec|- -- + ----|
|       2/  pi   pi*a\|    \  4    4*b /
|    sec |- -- + ----||                 
\        \  4    4*b //                 
$$\frac{2 \sec{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{\pi a}{4 b} - \frac{\pi}{4} \right)}}\right) \sec{\left(\frac{\pi a}{4 b} - \frac{\pi}{4} \right)}}$$
/       /pi*a\\ /       /pi*a\\
|1 - sin|----||*|1 + tan|----||
\       \2*b // \       \4*b //
-------------------------------
                /pi*a\         
         1 - tan|----|         
                \4*b /         
$$\frac{\left(1 - \sin{\left(\frac{\pi a}{2 b} \right)}\right) \left(\tan{\left(\frac{\pi a}{4 b} \right)} + 1\right)}{1 - \tan{\left(\frac{\pi a}{4 b} \right)}}$$
          /pi*a\  
     2*tan|----|  
          \4*b /  
1 + --------------
           2/pi*a\
    1 + tan |----|
            \4*b /
------------------
     /pi   pi*a\  
  tan|-- + ----|  
     \4    4*b /  
$$\frac{1 + \frac{2 \tan{\left(\frac{\pi a}{4 b} \right)}}{\tan^{2}{\left(\frac{\pi a}{4 b} \right)} + 1}}{\tan{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
      1       
--------------
   /pi   pi*a\
csc|-- - ----|
   \2    2*b /
$$\frac{1}{\csc{\left(- \frac{\pi a}{2 b} + \frac{\pi}{2} \right)}}$$
/        1    \    /pi   pi*a\
|1 + ---------|*csc|-- + ----|
|       /pi*a\|    \4    4*b /
|    csc|----||               
\       \2*b //               
------------------------------
           /pi   pi*a\        
        csc|-- - ----|        
           \4    4*b /        
$$\frac{\left(1 + \frac{1}{\csc{\left(\frac{\pi a}{2 b} \right)}}\right) \csc{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\csc{\left(- \frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
               /  pi   pi*a\          
          2*cos|- -- + ----|          
               \  4    4*b /          
--------------------------------------
/       2/  pi   pi*a\\               
|    cos |- -- + ----||               
|        \  4    4*b /|    /pi   pi*a\
|1 + -----------------|*cos|-- + ----|
|        2/pi   pi*a\ |    \4    4*b /
|     cos |-- + ----| |               
\         \4    4*b / /               
$$\frac{2 \cos{\left(\frac{\pi a}{4 b} - \frac{\pi}{4} \right)}}{\left(\frac{\cos^{2}{\left(\frac{\pi a}{4 b} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}} + 1\right) \cos{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
/          /pi*a\  \               
|     2*cot|----|  |               
|          \4*b /  |    /pi   pi*a\
|1 + --------------|*cot|-- + ----|
|           2/pi*a\|    \4    4*b /
|    1 + cot |----||               
\            \4*b //               
$$\left(1 + \frac{2 \cot{\left(\frac{\pi a}{4 b} \right)}}{\cot^{2}{\left(\frac{\pi a}{4 b} \right)} + 1}\right) \cot{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}$$
/       /  pi   pi*a\\    /pi   pi*a\
|1 + cos|- -- + ----||*cos|-- + ----|
\       \  2    2*b //    \4    4*b /
-------------------------------------
              /  pi   pi*a\          
           cos|- -- + ----|          
              \  4    4*b /          
$$\frac{\left(\cos{\left(\frac{\pi a}{2 b} - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\cos{\left(\frac{\pi a}{4 b} - \frac{\pi}{4} \right)}}$$
     1     
-----------
   /-pi*a \
sec|------|
   \ 2*b  /
$$\frac{1}{\sec{\left(- \frac{\pi a}{2 b} \right)}}$$
/       /pi*a\\ /       /pi*a\\
|1 - tan|----||*|1 + sin|----||
\       \4*b // \       \2*b //
-------------------------------
                /pi*a\         
         1 + tan|----|         
                \4*b /         
$$\frac{\left(1 - \tan{\left(\frac{\pi a}{4 b} \right)}\right) \left(\sin{\left(\frac{\pi a}{2 b} \right)} + 1\right)}{\tan{\left(\frac{\pi a}{4 b} \right)} + 1}$$
/       /pi*a\\    /pi   pi*a\
|1 + sin|----||*cot|-- + ----|
\       \2*b //    \4    4*b /
$$\left(\sin{\left(\frac{\pi a}{2 b} \right)} + 1\right) \cot{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}$$
       /pi*a\ 
1 + sin|----| 
       \2*b / 
--------------
   /pi   pi*a\
tan|-- + ----|
   \4    4*b /
$$\frac{\sin{\left(\frac{\pi a}{2 b} \right)} + 1}{\tan{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
           /       /pi*a\\          
         2*|1 + sin|----||          
           \       \2*b //          
------------------------------------
/         4/   /1    a \\\          
|    8*sin |pi*|- + ---|||          
|          \   \4   4*b//|    /pi*a\
|1 + --------------------|*cos|----|
|              /pi*a\    |    \2*b /
|       1 + cos|----|    |          
\              \ b  /    /          
$$\frac{2 \left(\sin{\left(\frac{\pi a}{2 b} \right)} + 1\right)}{\left(1 + \frac{8 \sin^{4}{\left(\pi \left(\frac{a}{4 b} + \frac{1}{4}\right) \right)}}{\cos{\left(\frac{\pi a}{b} \right)} + 1}\right) \cos{\left(\frac{\pi a}{2 b} \right)}}$$
   /pi   pi*a\
sin|-- + ----|
   \2    2*b /
$$\sin{\left(\frac{\pi a}{2 b} + \frac{\pi}{2} \right)}$$
       /pi   pi*a\ 
  2*tan|-- + ----| 
       \4    4*b / 
-------------------
       2/pi   pi*a\
1 + tan |-- + ----|
        \4    4*b /
$$\frac{2 \tan{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)} + 1}$$
/           1        \    /  pi   pi*a\
|1 + ----------------|*sec|- -- + ----|
|       /  pi   pi*a\|    \  4    4*b /
|    sec|- -- + ----||                 
\       \  2    2*b //                 
---------------------------------------
                /pi   pi*a\            
             sec|-- + ----|            
                \4    4*b /            
$$\frac{\left(1 + \frac{1}{\sec{\left(\frac{\pi a}{2 b} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{\pi a}{4 b} - \frac{\pi}{4} \right)}}{\sec{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
                 2                  
------------------------------------
/           1       \    /pi   pi*a\
|1 + ---------------|*tan|-- + ----|
|       2/pi   pi*a\|    \4    4*b /
|    tan |-- + ----||               
\        \4    4*b //               
$$\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}\right) \tan{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
   /pi*a\
cos|----|
   \2*b /
$$\cos{\left(\frac{\pi a}{2 b} \right)}$$
/       /pi*a\\    /pi*a\
|1 + sin|----||*cos|----|
\       \2*b //    \2*b /
-------------------------
         2/pi   pi*a\    
    2*sin |-- + ----|    
          \4    4*b /    
$$\frac{\left(\sin{\left(\frac{\pi a}{2 b} \right)} + 1\right) \cos{\left(\frac{\pi a}{2 b} \right)}}{2 \sin^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}$$
       2/pi*a\
1 - tan |----|
        \4*b /
--------------
       2/pi*a\
1 + tan |----|
        \4*b /
$$\frac{1 - \tan^{2}{\left(\frac{\pi a}{4 b} \right)}}{\tan^{2}{\left(\frac{\pi a}{4 b} \right)} + 1}$$
    1    
---------
   /pi*a\
sec|----|
   \2*b /
$$\frac{1}{\sec{\left(\frac{\pi a}{2 b} \right)}}$$
       /pi   pi*a\ 
  2*cot|-- + ----| 
       \4    4*b / 
-------------------
       2/pi   pi*a\
1 + cot |-- + ----|
        \4    4*b /
$$\frac{2 \cot{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)} + 1}$$
             2/pi   pi*a\        
        4*sin |-- + ----|        
              \4    4*b /        
---------------------------------
/         4/pi   pi*a\\          
|    4*sin |-- + ----||          
|          \4    4*b /|    /pi*a\
|1 + -----------------|*cos|----|
|           2/pi*a\   |    \2*b /
|        cos |----|   |          
\            \2*b /   /          
$$\frac{4 \sin^{2}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{\pi a}{4 b} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{\pi a}{2 b} \right)}} + 1\right) \cos{\left(\frac{\pi a}{2 b} \right)}}$$
   /-pi*a \
cos|------|
   \ 2*b  /
$$\cos{\left(- \frac{\pi a}{2 b} \right)}$$
cos(-pi*a/(2*b))