Answer
$\frac{cos~x}{1+sin~x}+\frac{1+sin~x}{cos~x}=2~sec~x$
Work Step by Step
$\frac{cos~x}{1+sin~x}+\frac{1+sin~x}{cos~x}=\frac{cos~x}{1+sin~x}\frac{cos~x}{cos~x}+\frac{1+sin~x}{cos~x}\frac{1+sin~x}{1+sin~x}=\frac{cos^2x+sin^2x+1+2~sin~x}{cos~x(1+sin~x)}=\frac{2(1+sin~x)}{cos~x(1+sin~x)}=\frac{2}{cos~x}=2~sec~x$
