## Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$

## Problem 612

Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.

Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.

**(a)** Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.

**(b)** Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.