Proving T_n(x) as Polynomials: Trig Homework Solution

[Dragonfall]

How do I show that

$$T_n(x)=\cos(n\arccos(x))$$

are actually polynomials of degree n?

If n=2 then

$$T_2(x)=x^2-(1-x^2)$$

but this seems to break down as n increases.

 

[StatusX]

So if x=cos(y), you want to show cos(ny) is a nth degree polynomial in cos(y). Induction is usually the first thing to try whenever there's an n knocking about.

 

[Office_Shredder]

It also definitely fails for n=1, but since you started with 2, maybe we're ignoring that

 

[d_leet]

It also definitely fails for n=1, but since you started with 2, maybe we're ignoring that

How does it fail for n=1? The function f(x)=x is a polynomial.