"Chubbychev" Polynomials Practice: identities

Since $T_n (\cos{\theta}) = \cos {(n\theta)}\quad \Rightarrow x = \cos {\theta}$

$\Rightarrow T_{90}(\cos{\theta}) = \cos{(90\theta)} = \frac {\sqrt{3}}{2} = \cos{\frac{\pi}{6}} \quad \Rightarrow \theta = \frac {\pi}{540}$

$\Rightarrow T_{360}(x) = T_{360}(\cos{\theta}) = \cos{\left( 360\theta \right)} = \cos{\left( \frac {360\pi}{540} \right)} = \cos{\left( \frac {2\pi}{3} \right)} = \boxed{-\frac{1}{2}}$

let $x=\cos(u)$

$T_n(\cos(u))=\cos(nu)$

Thus we have

$T_{90}(\cos(u))=\cos(90u)$

using the identity $\cos(2x)=2\cos^2(x)-1$

$T_{180}(\cos(u))=2(T_{90}(\cos(u)))^2-1=\frac{1}{2}$

using this identity once again

$T_{360}(\cos(u))=2(T_{180}(\cos(u)))^2-1=-0.5$