Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.

Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+\cdots+a_1 (x+x^{-1} )+a_0$$ on induction in $m$.


*

*$m=1$;

*$m=k$;

*$⊐n=k+1$. Than
$$x^{k+1}+x^{-k-1}=(x+x^{-1} )^{k+1} + (x+x^{-1} )^{k}+ (x+x^{-1} )^{k-1}+\cdots+(x+x^{-1} ).$$
I stuck on step 3. How to prove this expression?

$$\cos m \theta = T_m(\cos \theta)$$
with $T_m$ the Chebyshev polynomial of first kind, so, taking $x = e^{i\theta}$ 
$$x^{m} + x^{-m} = P_m(x+ x^{-1})$$
where $P_m(t) = 2 T_m(\frac{t}{2})$. For instance
$$x^{12} + x^{-12}= P_{12}(x+x^{-1})$$ 
where $P_{12}(t) = t^{12}-12 t^{10}+54 t^8-112 t^6+105 t^4-36 t^2+2$.
