I am stuck at something very trivial. I want to show that the Laplacian operator
$$ \frac{1}{r^2} \frac{\partial}{\partial r} \bigg(r^2 \frac{\partial f}{\partial r}\bigg)+ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \sin \theta \frac{\partial f}{\partial \theta} + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2}$$
is rotationally invariant. I can show it in cartesian coordinates but I want to show it spherical coordinates. Suppose I only look at a rotation $\theta \rightarrow \theta + \delta \theta$, and expand in powers of $\delta \theta$ keeping only terms linear in $\delta \theta$. If all goes well, then terms linear in $\delta \theta$ should vanish. For instance, assuming $f=f(r,\theta)$ the last term vanishes. The first term is invariant. But the second term gives a nonvanishing term
$$-\frac{1}{r^2} (\cot^2 \theta + 1) \frac{\partial f}{\partial \theta} \delta \theta.$$
Obviously I am making a mistake, but not in the calculation, probably it is conceptual in my strategy.
Any help would be appreciated. Thanks!