Answer
See below.
Work Step by Step
$LHS=\frac{1+sin\theta+cos\theta}{1+sin\theta -cos\theta}=\frac{1+sin\theta+cos\theta}{1+sin\theta -cos\theta}\times\frac{1+cos\theta}{1+cos\theta}=\frac{(1+sin\theta+cos\theta)(1+cos\theta)}{sin\theta+sin\theta cos\theta+1-cos^2\theta}=\frac{(1+sin\theta+cos\theta)(1+cos\theta)}{sin\theta+sin\theta cos\theta+sin^2\theta}=\frac{(1+sin\theta+cos\theta)(1+cos\theta)}{sin\theta(1+sin\theta+cos\theta)}=\frac{1+cos\theta}{sin\theta}=RHS$
