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

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.

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