Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$

HINT: $$\begin{align*} \frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x}&=\frac{\cos^2 x+(1+\sin x)^2}{\cos x(1+\sin x)}\\ &=\frac{\cos^2 x+\sin^2x+1+2\sin x}{\cos x(1+\sin x)}\;; \end{align*}$$

now use a familiar trig identity and find something to cancel.

Hint: if you put the two fractions over a common denominator you get a nice cancellation.