Answer
See below.
Work Step by Step
Step 1. To prove the given identify, we need to show $sin\theta(1+sin\theta+cos\theta)=(1+cos\theta)(1+sin\theta-cos\theta)$
Step 2. $LHS=sin\theta+sin^2\theta+sin\theta cos\theta$
Step 3. $RHS=1+sin\theta-cos\theta+cos\theta+sin\theta cos\theta-cos^2\theta=sin\theta+sin\theta cos\theta+sin^2\theta=LHS$
