Answer
See below.
Work Step by Step
$LHS=\frac{csc\theta+cot\theta}{sec\theta+tan\theta}
=\frac{csc\theta+cot\theta}{sec\theta+tan\theta}\times\frac{sec\theta-tan\theta}{sec\theta-tan\theta}
=\frac{(csc\theta+cot\theta)(sec\theta-tan\theta)}{sec^2\theta-tan^2\theta}
=(csc\theta+cot\theta)(sec\theta-tan\theta)$
$RHS=\frac{sec\theta-tan\theta}{csc\theta-cot\theta}
=\frac{sec\theta-tan\theta}{csc\theta-cot\theta}\times\frac{csc\theta+cot\theta}{csc\theta+cot\theta}
=\frac{(sec\theta-tan\theta)(csc\theta+cot\theta)}{csc^2\theta-cot^2\theta}
=(sec\theta-tan\theta)(csc\theta+cot\theta)=LHS$
