Question

`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec  theta)`

Answer 1

LHS= `tan theta/((1-cot theta))+ cot theta/((1-tan theta))`

      =`tan theta/((1-cos theta/sin theta)) + cot theta/((1-sin theta/cos theta))`

     =`(sin theta tan theta)/((sin theta- cos theta))+(cos theta cot theta)/((cos theta - sin theta))`

    =`(sin  theta  xx (sin theta) / (cos theta) cos theta xx (cos theta) / (sin theta))/((sin theta - cos theta))`

   =`((sin ^2 theta  cos ^2 theta)/(cos theta   sin theta))/((sin theta-cos theta))`

   =`( sin ^3 theta - cos ^3 theta)/(cos theta sin theta (sin theta - cos theta))`

 =` ((sin theta - cos theta)(sin ^2 theta + sin theta cos theta + cos ^2theta ))/(cos theta sin theta (sin theta- costheta))`

 =`(1+ sin theta cos theta)/(cos theta sin theta)`

 =`1/(cos theta sin theta)+(sin theta cos theta)/(cos theta  sin theta)`

  =`1/(cos theta sin theta)+ (sin theta cos theta)/(cos theta sin theta)`

  =`sectheta cosec  theta +1` 

  =`1+ sec theta  cosec  theta`

  =RHS