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

Therefore, we have:

`(\cos x)/(1+\sin x)-(\cos x)/(1-\sin x)=-2\tan x`