= Solution
For distinct positive integers $n,m$, use the product-to-sum identity in the <inner product>:
$$
\begin{aligned}
\langle\Psi_m|\Psi_n\rangle
&=\frac2a\int_0^a\sin\frac{m\pi x}{a}\sin\frac{n\pi x}{a}\,\mathrm dx\\
&=\frac1a\int_0^a\left[\cos\frac{(m-n)\pi x}{a}-\cos\frac{(m+n)\pi x}{a}\right]\,\mathrm dx\\
&=\left[\frac{\sin((m-n)\pi x/a)}{(m-n)\pi}-\frac{\sin((m+n)\pi x/a)}{(m+n)\pi}\right]_0^a=0.
\end{aligned}
$$
Every endpoint sine vanishes. For $m=n$, the normalization integral is one; the expression with a denominator $m-n$ should not be substituted directly. Thus \b[the normalized functions form an orthonormal set], $\langle\Psi_m|\Psi_n\rangle=\delta_{mn}$.
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