Chebyshev Polynomials - verifying

Question

How can I show the following properties of Chebyshev polynomials if I am only given the recursion formula and $$T_0(x)=1 \\ T_1(x)=x \\ T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$$

Verify

$$T_n(\cos\theta)=\cos(n\theta)\\ |T_n(x)| \le 1$$

Second one can easily be verified, but stuck on the first one.

Answer 1

Hint:

By direct substitution,

$$T_{n+1}(\cos\theta)=2\cos\theta\,T_n(\cos\theta)-T_{n-1}(\cos\theta)$$

becomes

$$\cos(n+1)\theta=2\cos\theta\cos n\theta-\cos(n-1)\theta.$$

Answer 2

Sketch:

• Use induction on $n$.

• The cases for $T_0$ and $T_1$ are direct.

• Use the product to sum formula (trig identity) for the inductive case.