Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$

Question

Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer.

Show that $$T_2 (x) = 2x^2 – 1.$$

My attempt:

$T_2 (x) = \cos (2 \arccos x)$

Because an identity for $\cos(ab)$ doesn't exist, I do not how to simplify further to get to the solution.

Answer 1

Hint. You have $$ \cos (2x)=\cos^2(x)-\sin^2(x)=2\cos^2(x)-1 $$ using $\sin^2x=1-\cos^2 x$ and $$ \cos (a+b)=\cos(a)\cos(b)-\sin(a)\sin(b) $$ with $a=b=x$.