Question 47
Suppose $P(x)$ is a polynomial with real coefficients satisfying the condition $P(\cos \theta + \sin \theta) = P(\cos \theta - \sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form
$$
P(x) = a_{0} + a_{1} (1 - x^{2})^{2} + a_{2} (1 - x^{2})^{4} + \cdots + a_{n} (1 - x^{2})^{2n}
$$
for some real numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ and nonnegative integer $n$.
$$
P(x) = a_{0} + a_{1} (1 - x^{2})^{2} + a_{2} (1 - x^{2})^{4} + \cdots + a_{n} (1 - x^{2})^{2n}
$$
for some real numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ and nonnegative integer $n$.