Expressing an inverse trig function?

Question

I just need a little help with this question:

"Express cos$y$ in terms of cos $y/2$ and hence show that tan$^{-1} sqrt[(1-x)/(1+x)] = 1/2$ cos$^{-1}x$, for $0<x<1$."

I can do the first part, where I express cos$y$ in terms of cos $y/2$, but how do I do the second part of the question?

Thanks.

Answer 1

You know then that $\cos(2x) = \cos^2x - \sin^2x$. You also know that $\cos^2x + \sin^2x = 1$. Then dividing by $1$ does not alter anything and so:
$$\cos(2x) = \dfrac{\cos^2x - \sin^2x}{\cos^2x + \sin^2x}$$
Divide numerator and denominator by $\cos^2x$; replace $\dfrac{\sin x}{\cos x}$ by ???

I am sure you can take from here.