Question

Prove the following identities:

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

Answer 1

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

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

      =`((sin theta + 1 )^2 - cos^2 theta)/((sin theta + cos theta )^2 -1^2)`

       =`(sin^2 theta +1+2 sin theta - cos^2 theta)/(sin^2 + cos^2 theta+2 sin theta  cos theta -1)`

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

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

      =`(2 sin theta (1+ sin theta))/(2 sin theta cos theta)`

      =`(1+sin theta)/cos theta`

      = RHS