Question

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of the answer.

Answer 1

Let's simplify the expression:

$\begin{gathered} \frac{\cos x}{1+\sin x}-\frac{\cos x}{1-\sin x}=\frac{\cos x(1-\sin x)-\cos x(1+\sin x)}{(1+\sin x)(1-\sin x)} \\ =\frac{\cos x(1-\sin x-1-\sin x)}{1-\sin^2x} \\ =\frac{-2\sin x\cos x}{1-\sin^2x} \\ \text{ Now we use the pythagorean identity }\sin^2x+\cos^2x=1: \\ \frac{-2\sin x\cos x}{1-\sin^2x}=-\frac{2\sin x\cos x}{\cos^2x} \\ =-\frac{2\sin x}{\cos x} \\ \text{ Finally we need to remember that }\tan x=\frac{\sin x}{\cos x},\text{ then:} \\ -\frac{2\sin x}{\cos x}=-2\tan x \end{gathered}$

Therefore, we have:

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