= Solution
Multiplication by $x^2$ gives the <Sturm-Liouville form>
$$
-(x^2u')'=\lambda x^2u,
$$
so the weight is $w(x)=x^2$. Use the weighted <inner product> $\langle f,g\rangle_w=\int_0^{\pi/2}x^2\overline f g\,dx$ and the <Sturm-Liouville operator> $L=-x^{-2}(x^2d/dx)'$. <Integration by parts> gives its boundary form:
$$
\langle f,Lg\rangle_w-\langle Lf,g\rangle_w
=\left[x^2(\overline {f'}g-\overline f g')\right]_0^{\pi/2}=0.
$$
The upper endpoint term vanishes by the <Dirichlet boundary conditions>, and the lower one vanishes for the regular solutions. More precisely, the <radial Dirichlet spectrum in three dimensions> identifies $L$ under the <unitary transformation> $v=xu$ with $-v''$ on $(0,\pi/2)$ with zero endpoint values; this specifies the <self-adjoint> realization at the singular endpoint as well as proving the formal identity.
For <eigenfunctions> $u_a,u_b$ with different <eigenvalues>, the same identity gives $(\lambda_b-\lambda_a)\langle u_a,u_b\rangle_w=0$. Taking real <eigenfunctions> proves the required <orthogonality of Sturm-Liouville eigenfunctions>.
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