obtain $y=\cos2\alpha$ knowing $x=\tan\alpha$

Question

obtain $y=\cos2\alpha$ knowing $x=\tan\alpha$

So obviously I tried the double angle identity to see where it gets me:

$$\cos2\alpha = \cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha$$ $$1-2\sin^2\alpha = 1-2\tan^2\alpha\cos^2\alpha=1-2x^2\cos^2\alpha$$

I'm not sure how to continue in order show $y$ with $x$ without the cos or sin functions..

any tips or hints?

---

I'm not a student yet, this is one of the Tel-Aviv university perparation exercises for the entry test so this shouln't too hard.

Answer 1

$$y=\cos2\alpha=2\cos^2\alpha-1=\frac{2\cos^2\alpha}{\cos^2\alpha+\sin^2\alpha}-1=\frac2{1+\tan^2\alpha}-1=\frac2{1+x^2}-1=\frac{1-x^2}{1+x^2}.$$