Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$ I am fairly good at solving trig equations yet this one equation has me stumped. I've been trying very hard but was unable to solve it. Can anyone help please? Thank you.
$$\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$$
solve for $x$ in the range of $[-2\pi, 2\pi]$
I do know we have to do a difference of squares, yet after that, I don't know what to do and I get lost.
Thank you.

You have a fraction and its reciprocal adding to $2$: $\lambda + 1/\lambda=2$. Only solution is $\lambda=1$, as @J.M. has noted. So $\cos x = 1+\sin x$. Draw the graphs of the two sides of the equation and see that there are five intersections in the $x$-interval $[-2\pi,2\pi]$, but the two at $x=-\pi/2$ and $x=3\pi/2$ are no good, ’cause there you get $0/0$ occurring in the original problem.
