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Expression simplification/ Least common denominator/ tan(pi/4+x/2)

Least common denominator tan(pi/4+x/2)

An expression to simplify:

The solution

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