Simplify  
cotθ
1 - tanθ
  +  
tanθ
1 - cotθ
 


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Answer:

1 + tanθ + cotθ

Step by Step Explanation:
  1. We know that, tanθ =  
    sinθ
    cosθ
     , cotθ =  
    cosθ
    sinθ
     
  2. Now,  
    cotθ
    1 - tanθ
      +  
    tanθ
    1 - cotθ
      can be simplified as:

     
    cotθ
    1 - tanθ
      +  
    tanθ
    1 - cotθ
     
    =  
     
    cosθ
    sinθ
     
    1 -  
    sinθ
    cosθ
     
      +  
     
    sinθ
    cosθ
     
    1 -  
    cosθ
    sinθ
     
     
    =  
     
    cosθ
    sinθ
     
     
    cosθ - sinθ
    cosθ
     
      +  
     
    sinθ
    cosθ
     
     
    sinθ - cosθ
    sinθ
     
     
    =  
    cos2θ
    sinθ(cosθ - sinθ)
      +  
    sin2θ
    cosθ(sinθ - cosθ)
     
    =  
    cos3θ - sin3θ
    sinθ.cosθ(cosθ - sinθ)
     
    =  
    (cosθ - sinθ)(cos2θ + cosθ.sinθ + sin2θ)
    sinθ.cosθ(cosθ - sinθ)
     
    =  
    cos2θ + cosθ.sinθ + sin2θ
    sinθ.cosθ
     
    =  
    cos2θ
    cosθ.sinθ
      +  
    cosθ.sinθ
    cosθ.sinθ
      +  
    sin2θ
    cosθ.sinθ
     
    =  
    cosθ
    sinθ
      + 1 +  
    sinθ
    cosθ
     
    = cotθ + 1 + tanθ
    = 1 + tanθ + cotθ

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