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Expression simplification/ Fraction Decomposition into the simple/ -tan(-1/2+x)/(-1+tan(-1/2+x)^2)

How do you -tan(-1/2+x)/(-1+tan(-1/2+x)^2) in partial fractions?

An expression to simplify:

The solution

You have entered [src] 
  -tan(-1/2 + x)   
-------------------
        2          
-1 + tan (-1/2 + x)
$$\frac{\left(-1\right) \tan{\left(x - \frac{1}{2} \right)}}{\tan^{2}{\left(x - \frac{1}{2} \right)} - 1}$$ 
(-tan(-1/2 + x))/(-1 + tan(-1/2 + x)^2)
Fraction decomposition [src] 
-1/(2*(1 + tan(-1/2 + x))) - 1/(2*(-1 + tan(-1/2 + x)))
$$- \frac{1}{2 \left(\tan{\left(x - \frac{1}{2} \right)} + 1\right)} - \frac{1}{2 \left(\tan{\left(x - \frac{1}{2} \right)} - 1\right)}$$ 
            1                       1           
- --------------------- - ----------------------
  2*(1 + tan(-1/2 + x))   2*(-1 + tan(-1/2 + x))
General simplification [src] 
  -tan(-1/2 + x)   
-------------------
        2          
-1 + tan (-1/2 + x)
$$- \frac{\tan{\left(x - \frac{1}{2} \right)}}{\tan^{2}{\left(x - \frac{1}{2} \right)} - 1}$$ 
-tan(-1/2 + x)/(-1 + tan(-1/2 + x)^2)
Common denominator [src] 
  -tan(-1/2 + x)   
-------------------
        2          
-1 + tan (-1/2 + x)
$$- \frac{\tan{\left(x - \frac{1}{2} \right)}}{\tan^{2}{\left(x - \frac{1}{2} \right)} - 1}$$ 
-tan(-1/2 + x)/(-1 + tan(-1/2 + x)^2)
Numerical answer [src] 
-tan(-1/2 + x)/(-1.0 + tan(-1/2 + x)^2)
-tan(-1/2 + x)/(-1.0 + tan(-1/2 + x)^2)
Combinatorics [src] 
            -tan(-1/2 + x)              
----------------------------------------
(1 + tan(-1/2 + x))*(-1 + tan(-1/2 + x))
$$- \frac{\tan{\left(x - \frac{1}{2} \right)}}{\left(\tan{\left(x - \frac{1}{2} \right)} - 1\right) \left(\tan{\left(x - \frac{1}{2} \right)} + 1\right)}$$ 
-tan(-1/2 + x)/((1 + tan(-1/2 + x))*(-1 + tan(-1/2 + x)))
Rational denominator [src] 
  -tan(-1/2 + x)   
-------------------
        2          
-1 + tan (-1/2 + x)
$$- \frac{\tan{\left(x - \frac{1}{2} \right)}}{\tan^{2}{\left(x - \frac{1}{2} \right)} - 1}$$ 
-tan(-1/2 + x)/(-1 + tan(-1/2 + x)^2)
Combining rational expressions [src] 
      /-1 + 2*x\   
  -tan|--------|   
      \   2    /   
-------------------
        2/-1 + 2*x\
-1 + tan |--------|
         \   2    /
$$- \frac{\tan{\left(\frac{2 x - 1}{2} \right)}}{\tan^{2}{\left(\frac{2 x - 1}{2} \right)} - 1}$$ 
-tan((-1 + 2*x)/2)/(-1 + tan((-1 + 2*x)/2)^2)
Powers [src] 
  -tan(-1/2 + x)   
-------------------
        2          
-1 + tan (-1/2 + x)
$$- \frac{\tan{\left(x - \frac{1}{2} \right)}}{\tan^{2}{\left(x - \frac{1}{2} \right)} - 1}$$ 
                     /   I*(-1/2 + x)    I*(1/2 - x)\                  
                  -I*\- e             + e           /                  
-----------------------------------------------------------------------
/                                     2\                               
|     /   I*(-1/2 + x)    I*(1/2 - x)\ |                               
|     \- e             + e           / | / I*(1/2 - x)    I*(-1/2 + x)\
|-1 - ---------------------------------|*\e            + e            /
|                                    2 |                               
|      / I*(1/2 - x)    I*(-1/2 + x)\  |                               
\      \e            + e            /  /
$$- \frac{i \left(e^{i \left(\frac{1}{2} - x\right)} - e^{i \left(x - \frac{1}{2}\right)}\right)}{\left(- \frac{\left(e^{i \left(\frac{1}{2} - x\right)} - e^{i \left(x - \frac{1}{2}\right)}\right)^{2}}{\left(e^{i \left(\frac{1}{2} - x\right)} + e^{i \left(x - \frac{1}{2}\right)}\right)^{2}} - 1\right) \left(e^{i \left(\frac{1}{2} - x\right)} + e^{i \left(x - \frac{1}{2}\right)}\right)}$$ 
-i*(-exp(i*(-1/2 + x)) + exp(i*(1/2 - x)))/((-1 - (-exp(i*(-1/2 + x)) + exp(i*(1/2 - x)))^2/(exp(i*(1/2 - x)) + exp(i*(-1/2 + x)))^2)*(exp(i*(1/2 - x)) + exp(i*(-1/2 + x))))
Assemble expression [src] 
  -tan(-1/2 + x)   
-------------------
        2          
-1 + tan (-1/2 + x)
$$- \frac{\tan{\left(x - \frac{1}{2} \right)}}{\tan^{2}{\left(x - \frac{1}{2} \right)} - 1}$$ 
-tan(-1/2 + x)/(-1 + tan(-1/2 + x)^2)
Expand expression [src] 
                                                                                                                                          tan(1/2)                                                                                                                                                                                                                                                                                        tan(x)                                                                                                                                           
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                        2                                            2                                                         3                                           3                                           2         2                                                                                     2                                            2                                                         3                                           3                                           2         2                                                          
                     tan (1/2)                                    tan (x)                                                   tan (1/2)*tan(x)                            tan (x)*tan(1/2)                          2*tan (1/2)*tan (x)                          2*tan(1/2)*tan(x)                                    tan (1/2)                                    tan (x)                                                   tan (1/2)*tan(x)                            tan (x)*tan(1/2)                          2*tan (1/2)*tan (x)                          2*tan(1/2)*tan(x)            
-1 + ----------------------------------------- + ----------------------------------------- - tan(1/2)*tan(x) + ----------------------------------------- + ----------------------------------------- - ----------------------------------------- - -----------------------------------------   -1 + ----------------------------------------- + ----------------------------------------- - tan(1/2)*tan(x) + ----------------------------------------- + ----------------------------------------- - ----------------------------------------- - -----------------------------------------
            2         2                                 2         2                                                   2         2                                 2         2                                 2         2                                 2         2                                      2         2                                 2         2                                                   2         2                                 2         2                                 2         2                                 2         2                       
     1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)                     1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)        1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)                     1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)   1 + tan (1/2)*tan (x) + 2*tan(1/2)*tan(x)
$$- \frac{\tan{\left(x \right)}}{- \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} - 1 + \frac{\tan{\left(\frac{1}{2} \right)} \tan^{3}{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} - \frac{2 \tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{2}{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} - \frac{2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{3}{\left(\frac{1}{2} \right)} \tan{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1}} + \frac{\tan{\left(\frac{1}{2} \right)}}{- \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} - 1 + \frac{\tan{\left(\frac{1}{2} \right)} \tan^{3}{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} - \frac{2 \tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{2}{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} - \frac{2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{3}{\left(\frac{1}{2} \right)} \tan{\left(x \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} \tan^{2}{\left(x \right)} + 2 \tan{\left(\frac{1}{2} \right)} \tan{\left(x \right)} + 1}}$$ 
tan(1/2)/(-1 + tan(1/2)^2/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) + tan(x)^2/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) - tan(1/2)*tan(x) + tan(1/2)^3*tan(x)/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) + tan(x)^3*tan(1/2)/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) - 2*tan(1/2)^2*tan(x)^2/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) - 2*tan(1/2)*tan(x)/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x))) - tan(x)/(-1 + tan(1/2)^2/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) + tan(x)^2/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) - tan(1/2)*tan(x) + tan(1/2)^3*tan(x)/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) + tan(x)^3*tan(1/2)/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) - 2*tan(1/2)^2*tan(x)^2/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)) - 2*tan(1/2)*tan(x)/(1 + tan(1/2)^2*tan(x)^2 + 2*tan(1/2)*tan(x)))
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