Proving that two function coordinates of a parametric curve equals 1

Question

I am having difficulty with this question,

Note: This is not homework, It is from a practice test that I am using to study

Consider the curve:

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

The question is to prove that $x^2(t) + y^2(t) = 1$ for all $t$

I have no clue where to start, I think it has something to do with the unit circle, but I'm not sure

Also, to find the slope of a parametric curve, Am i correct in assuming that you simply differentiate both of them, plug in the x and y, and use rise over run?

Thanks!

Answer 1

Hint:

Take t=cos $\theta$. You will get x=$\cos2\theta$ and y=$\sin 2\theta$. Eliminate $\theta$ by using $$\cos^2 \theta+ \sin ^2\theta=1$$