How does $\frac{1-\cos x}{1+\cos x}$ become $\frac{2\sin^2(x/2)}{2\cos^2(x/2)}$?

Question

I am new to calculus.

How?

thanks in advance

Answer 1

$1- \cos x=1-\cos(\frac{x}{2}+\frac{x}{2})=1-(\cos^2(\frac{x}{2})-\sin^2(\frac{x}{2}))=1-(1-\sin^2(\frac{x}{2}))+\sin^2(\frac{x}{2})=2 \sin^2(\frac{x}{2}).$

Answer 2

Use that:

$$\cos(2\theta)\equiv 1-2\sin^2(\theta)\equiv2\cos^2(\theta)-1$$