Verify $\cos(x)=\frac{1-t^2}{1+t^2}$ with $t=\tan(\frac{x}{2})$

Question

I was requested to verify, with $t=\tan(\frac{x}{2})$, the following identity:

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

I'm quite rusty on my trigonometry, and hasn't been able to found the proof of this. I'm sure there may be some trigonometric property I should know to simplify the work. Could someone hint me or altotegher tell me how to solve this problem? I tried to simplify the RHS looking to get $\cos(x)$ out of it but failed.

Answer 1

Write $y=x/2$. Then, multiplying by $\cos^2y$ on top and bottom, $$\frac{1-\tan^2y}{1+\tan^2y}=\frac{\cos^2y-\sin^2y}{\cos^2y+\sin^2y}=\frac{\cos2y}1=\cos x$$ The denominator simplifies by the Pythagorean identity $\cos^2x+\sin^2x=1$ and the numerator simplifies by $\cos2x=\cos^2x-\sin^2x$.