Answer
See below
Work Step by Step
Obtain $=\int ^\pi_0 \cos x\sin x dx\\
=\int ^\pi_0 \frac{\sin 2x}{2}dx\\
=(-\frac{\cos 2x}{4})|^\pi_0\\
=(-\frac{\cos 2x}{4})-(-\frac{\cos 0}{4})\\
=(-\frac{1}{4})-(-\frac{1}{4})\\
=0$
The functions $f(x)=\cos x$ and $g(x)=\sin x$ are an orthogonal functions on $C[0,\pi]$. Hence, $\{\cos x,\sin x\}$ is an orthogonal basis for span $\{\cos x,\sin x\}$
