Deriving the recurrence relation for Chebyshev polynomials using law of cosines?

Question

I am trying to derive the recurrence relation in the Chebyshev polynomial using the following recurrence relation:

$\cos((n+1)\cos^{-1}x)$ $= x\cos(n\cos^{-1}x) $ - $\sin(n\cos^{-1}x)\sin(\cos^{-1}x)$

I don't know how to proceed from here.

Answer 1

Hint:

Start from this factorisation formula: $$\cos(n+1)\theta +\cos(n-1)\theta=2\cos \theta\cos n\theta. $$