Write the given expression as an algebraic expression in x. cos(2cos^−1 x)

Write the given expression as an algebraic expression in x.
cos(2 cos^−1 x)

I'm not really sure how to get this one. I've made 6 attempts and they've all been wrong. I know this one probably isn't that hard, I'm just not approaching it right.

y = cos(2cosˉ¹(x))

Let's make a substitution that will clarify this a bit:

Let θ = cosˉ¹(x) then use the double angle identity to write

y = cos(2θ) = cos²θ -sin²θ = 2cos²θ -1

Now, since θ is the angle whose cosine is x we can substitute:

y = 2x² -1

cos(2cosˉ¹(x)) = 2x² -1