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