Answer
The identity is verified.
$\frac{1+sin~\theta}{cos~\theta}+\frac{cos~\theta}{1+sin~\theta}=2~sec~\theta$
Work Step by Step
$\frac{1+sin~\theta}{cos~\theta}+\frac{cos~\theta}{1+sin~\theta}=\frac{1+sin~\theta}{cos~\theta}\frac{1+sin~\theta}{1+sin~\theta}+\frac{cos~\theta}{1+sin~\theta}\frac{cos~\theta}{cos~\theta}=\frac{1+2~sin~\theta+sin^2\theta+cos^2\theta}{(1+sin~\theta)cos~\theta}=\frac{2+2~sin~\theta}{(1+sin~\theta)cos~\theta}=\frac{2(1+sin~\theta)}{(1+sin~\theta)cos~\theta}=2\frac{1}{cos~\theta}=2~sec~\theta$
