Existence of Chebyshev Polynomials of the Second Kind

Theorem
There exists a Chebyshev polynomial of the second kind for all natural numbers $n$.

Proof
For $n = 0$:


 * $\map {U_0} x = 1$, $U_0 \in \Bbb P$

For $n = 1$:


 * $\map {U_1} x = 2x$, $U_1 \in \Bbb P$

Assume $\map {U_n} {\cos \theta}\sin \theta = \map \sin { \paren {n+1} \theta }$.


 * $\map {U_{n+1} } x = 2x \map {U_n} x - \map {U_{n - 1} } x$, $U_{n+1} \in \Bbb P$

By the Second Principle of Mathematical Induction, $U_n \in \Bbb P$ for all natural numbers $n$.