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