Use the first kind Chebyshev polynomial $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ to show how the leading coefficient is always positive $(1, 2, 4, 8, 16, 32...)$ when $n\geq 1$ using proof by induction
$$T_0(x) = 1$$
$$T_1(x) = x $$
$$T_2(x) = 2x^2 − 1 $$
$$T_3(x) = 4x^3 − 3x $$
$$T_4(x) = 8x^4 − 8x^2 + 1 $$
We show by induction that for every $n\geq 1$ $lc(T_n)=2^{n-1}$ (where $lc$ stands for leading coefficient). For $n=1$ the thesis is true.
Suppose we have the thesis for every $m< n+1$. We know that $T_{n+1}(x)=2xT_n(x)-T_{n-1}$. Since $deg(T_k)=k$ we know that the leading term of $T_{n+1}$, $lt(T_{n+1})$, will be $2xlt(T_n)=2\cdot2^{n-1} x^n$, where the last equality follows by induction. Hence $lc(T_{n+1})=2^n$, as wanted.