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Least common denominator:
(z*(z^2+6*z+1)-(4*z*(z+1)))/(z-1)^3+z*(z-1)/(z+1)^3
(z^5-5*z^3+10*z-10/z+1/5*z^3-1/z^5)*(z^3-3*z+3/z-1/z^3)
((y-9)/(y-8))*((y^2-64)/(y^2-16*y+64))
sin((x-y)/2)-sin((x+y)/2)
Factor polynomial:
x^2-49
-y^2+y
x^4+81
x^4-x^2
Factor squared:
x^2+4*x+5
x^2+x-4
p^4+9*p^2+5
y^4-y^2+9
How do you in partial fractions?:
-1/(2*x^2)
(pi*a^2+pi/4)/(pi*a^2+pi/2)
2432902008176640000/(-2+x)^21
-x^2/(x-2)^2+2*x/(x-2)
Identical expressions
sin((x-y)/ two)-sin((x+y)/ two)
sinus of ((x minus y) divide by 2) minus sinus of ((x plus y) divide by 2)
sinus of ((x minus y) divide by two) minus sinus of ((x plus y) divide by two)
sinx-y/2-sinx+y/2
sin((x-y) divide by 2)-sin((x+y) divide by 2)
Similar expressions
sin((x+y)/2)-sin((x+y)/2)
sin((x-y)/2)+sin((x+y)/2)
sin((x-y)/2)-sin((x-y)/2)

Expression simplification/ Least common denominator/ sin((x-y)/2)-sin((x+y)/2)

Least common denominator sin((x-y)/2)-sin((x+y)/2)

The solution

You have entered [src]

   /x - y\      /x + y\
sin|-----| - sin|-----|
   \  2  /      \  2  /

$$\sin{\left(\frac{x - y}{2} \right)} - \sin{\left(\frac{x + y}{2} \right)}$$

sin((x - y)/2) - sin((x + y)/2)

General simplification [src]

      /x\    /y\
-2*cos|-|*sin|-|
      \2/    \2/

$$- 2 \sin{\left(\frac{y}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

-2*cos(x/2)*sin(y/2)

Numerical answer [src]

-sin((x + y)/2) + sin((x - y)/2)

-sin((x + y)/2) + sin((x - y)/2)

Common denominator [src]

     /x   y\      /x   y\
- sin|- + -| + sin|- - -|
     \2   2/      \2   2/

$$\sin{\left(\frac{x}{2} - \frac{y}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{y}{2} \right)}$$

-sin(x/2 + y/2) + sin(x/2 - y/2)

Powers [src]

  /     /  x   y\      /x   y\\     /     /y   x\      /x   y\\
  |   I*|- - - -|    I*|- + -||     |   I*|- - -|    I*|- - -||
  |     \  2   2/      \2   2/|     |     \2   2/      \2   2/|
I*\- e            + e         /   I*\- e          + e         /
------------------------------- - -----------------------------
               2                                2

$$\frac{i \left(- e^{i \left(- \frac{x}{2} - \frac{y}{2}\right)} + e^{i \left(\frac{x}{2} + \frac{y}{2}\right)}\right)}{2} - \frac{i \left(- e^{i \left(- \frac{x}{2} + \frac{y}{2}\right)} + e^{i \left(\frac{x}{2} - \frac{y}{2}\right)}\right)}{2}$$

     /x   y\      /x   y\
- sin|- + -| + sin|- - -|
     \2   2/      \2   2/

$$\sin{\left(\frac{x}{2} - \frac{y}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{y}{2} \right)}$$

-sin(x/2 + y/2) + sin(x/2 - y/2)

Combining rational expressions [src]

     /x   y\      /x   y\
- sin|- + -| + sin|- - -|
     \2   2/      \2   2/

$$\sin{\left(\frac{x}{2} - \frac{y}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{y}{2} \right)}$$

-sin(x/2 + y/2) + sin(x/2 - y/2)

Combinatorics [src]

     /x   y\      /x   y\
- sin|- + -| + sin|- - -|
     \2   2/      \2   2/

$$\sin{\left(\frac{x}{2} - \frac{y}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{y}{2} \right)}$$

-sin(x/2 + y/2) + sin(x/2 - y/2)

Rational denominator [src]

     /x   y\      /x   y\
- sin|- + -| + sin|- - -|
     \2   2/      \2   2/

$$\sin{\left(\frac{x}{2} - \frac{y}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{y}{2} \right)}$$

-sin(x/2 + y/2) + sin(x/2 - y/2)

Expand expression [src]

      /x\    /y\
-2*cos|-|*sin|-|
      \2/    \2/

$$- 2 \sin{\left(\frac{y}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

-2*cos(x/2)*sin(y/2)

Trigonometric part [src]

  /         2/x\\    /y\    /y\
4*|1 - 2*cos |-||*cos|-|*sin|-|
  \          \4//    \4/    \4/

$$4 \left(1 - 2 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin{\left(\frac{y}{4} \right)} \cos{\left(\frac{y}{4} \right)}$$

       -2         
------------------
   /y\    /pi   x\
csc|-|*csc|-- - -|
   \2/    \2    2/

$$- \frac{2}{\csc{\left(\frac{y}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$

         /x   y\           /  y   x\ 
    2*tan|- + -|      2*tan|- - + -| 
         \4   4/           \  4   4/ 
- --------------- + -----------------
         2/x   y\          2/  y   x\
  1 + tan |- + -|   1 + tan |- - + -|
          \4   4/           \  4   4/

$$- \frac{2 \tan{\left(\frac{x}{4} + \frac{y}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} + \frac{y}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{4} - \frac{y}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} - \frac{y}{4} \right)} + 1}$$

         /x   y\           /  y   x\ 
    2*cot|- + -|      2*cot|- - + -| 
         \4   4/           \  4   4/ 
- --------------- + -----------------
         2/x   y\          2/  y   x\
  1 + cot |- + -|   1 + cot |- - + -|
          \4   4/           \  4   4/

$$- \frac{2 \cot{\left(\frac{x}{4} + \frac{y}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} + \frac{y}{4} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{4} - \frac{y}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} - \frac{y}{4} \right)} + 1}$$

                /                   2\       
                |      /       2/x\\ |       
                |    2*|1 - tan |-|| |       
  /       2/y\\ |      \        \8// |    /y\
8*|1 - tan |-||*|1 - ----------------|*tan|-|
  \        \8// |                  2 |    \8/
                |     /       2/x\\  |       
                |     |1 + tan |-||  |       
                \     \        \8//  /       
---------------------------------------------
                             2               
                /       2/y\\                
                |1 + tan |-||                
                \        \8//

$$\frac{8 \left(1 - \tan^{2}{\left(\frac{y}{8} \right)}\right) \left(- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + 1\right) \tan{\left(\frac{y}{8} \right)}}{\left(\tan^{2}{\left(\frac{y}{8} \right)} + 1\right)^{2}}$$

      /y\    /pi   x\
-2*sin|-|*sin|-- + -|
      \2/    \2    2/

$$- 2 \sin{\left(\frac{y}{2} \right)} \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}$$

       -2         
------------------
   /x\    /y   pi\
sec|-|*sec|- - --|
   \2/    \2   2 /

$$- \frac{2}{\sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}$$

      /x\    /y   pi\
-2*cos|-|*cos|- - --|
      \2/    \2   2 /

$$- 2 \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}$$

  /         2/x\\    /y\    /  pi   y\
4*|1 - 2*cos |-||*cos|-|*cos|- -- + -|
  \          \4//    \4/    \  2    4/

$$4 \left(1 - 2 \cos^{2}{\left(\frac{x}{4} \right)}\right) \cos{\left(\frac{y}{4} \right)} \cos{\left(\frac{y}{4} - \frac{\pi}{2} \right)}$$

  /         2      \
4*|1 - ------------|
  |       2/pi   x\|
  |    csc |-- - -||
  \        \2    4//
--------------------
    /y\    /pi   y\ 
 csc|-|*csc|-- - -| 
    \4/    \2    4/

$$\frac{4 \left(1 - \frac{2}{\csc^{2}{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}}\right)}{\csc{\left(\frac{y}{4} \right)} \csc{\left(- \frac{y}{4} + \frac{\pi}{2} \right)}}$$

  /                    2\                      
  |      /        2/x\\ |                      
  |    2*|-1 + cot |-|| |                      
  |      \         \8// | /        2/y\\    /y\
8*|1 - -----------------|*|-1 + cot |-||*cot|-|
  |                   2 | \         \8//    \8/
  |      /       2/x\\  |                      
  |      |1 + cot |-||  |                      
  \      \        \8//  /                      
-----------------------------------------------
                              2                
                 /       2/y\\                 
                 |1 + cot |-||                 
                 \        \8//

$$\frac{8 \left(- \frac{2 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + 1\right) \left(\cot^{2}{\left(\frac{y}{8} \right)} - 1\right) \cot{\left(\frac{y}{8} \right)}}{\left(\cot^{2}{\left(\frac{y}{8} \right)} + 1\right)^{2}}$$

     /        2/x\\    /y\ 
  -4*|-1 + cot |-||*cot|-| 
     \         \4//    \4/ 
---------------------------
/       2/x\\ /       2/y\\
|1 + cot |-||*|1 + cot |-||
\        \4// \        \4//

$$- \frac{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \cot{\left(\frac{y}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{y}{4} \right)} + 1\right)}$$

  /         2/pi   x\\    /y\    /pi   y\
4*|1 - 2*sin |-- + -||*sin|-|*sin|-- + -|
  \          \2    4//    \4/    \2    4/

$$4 \left(1 - 2 \sin^{2}{\left(\frac{x}{4} + \frac{\pi}{2} \right)}\right) \sin{\left(\frac{y}{4} \right)} \sin{\left(\frac{y}{4} + \frac{\pi}{2} \right)}$$

     /x   y   pi\      /x   pi   y\
- cos|- + - - --| + cos|- - -- - -|
     \2   2   2 /      \2   2    2/

$$\cos{\left(\frac{x}{2} - \frac{y}{2} - \frac{\pi}{2} \right)} - \cos{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}$$

      /x\    /y\
-2*cos|-|*sin|-|
      \2/    \2/

$$- 2 \sin{\left(\frac{y}{2} \right)} \cos{\left(\frac{x}{2} \right)}$$

     /       2/x\\    /y\  
  -4*|1 - tan |-||*tan|-|  
     \        \4//    \4/  
---------------------------
/       2/x\\ /       2/y\\
|1 + tan |-||*|1 + tan |-||
\        \4// \        \4//

$$- \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \tan{\left(\frac{y}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{4} \right)} + 1\right)}$$

     /x   y\      /x   y\
- sin|- + -| + sin|- - -|
     \2   2/      \2   2/

$$\sin{\left(\frac{x}{2} - \frac{y}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{y}{2} \right)}$$

       1                 1       
--------------- - ---------------
   /x   pi   y\      /x   y   pi\
sec|- - -- - -|   sec|- + - - --|
   \2   2    2/      \2   2   2 /

$$- \frac{1}{\sec{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(\frac{x}{2} - \frac{y}{2} - \frac{\pi}{2} \right)}}$$

    /       2   \   
  4*|1 - -------|   
    |       2/x\|   
    |    sec |-||   
    \        \4//   
--------------------
   /y\    /  pi   y\
sec|-|*sec|- -- + -|
   \4/    \  2    4/

$$\frac{4 \left(1 - \frac{2}{\sec^{2}{\left(\frac{x}{4} \right)}}\right)}{\sec{\left(\frac{y}{4} \right)} \sec{\left(\frac{y}{4} - \frac{\pi}{2} \right)}}$$

    1            1     
---------- - ----------
   /x   y\      /x   y\
csc|- - -|   csc|- + -|
   \2   2/      \2   2/

$$- \frac{1}{\csc{\left(\frac{x}{2} + \frac{y}{2} \right)}} + \frac{1}{\csc{\left(\frac{x}{2} - \frac{y}{2} \right)}}$$

1/csc(x/2 - y/2) - 1/csc(x/2 + y/2)

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How to use it?

Identical expressions

Similar expressions

Least common denominator sin((x-y)/2)-sin((x+y)/2)

The solution