In R. Shankar's "Principals of Quantum Mechanics" I've been asked, and have, proven that $$\delta(\mathrm f(x)) = \sum_i \frac{\delta(x-x_i)}{\left|\mathrm f'(x_i)\right|}$$ where $\mathrm f(x_i)=0$ for all $i$. So, for example: $$\delta(\sin x) = \sum_{n \in \mathbb Z} \frac{\delta(x-\pi n)}{\left|\cos(\pi n)\right|} = \sum_{n \in \mathbb Z} \frac{\delta(x-\pi n)}{\left|(-1)^n\right|} =\sum_{n \in \mathbb Z} \delta(x-\pi n)$$ That's all very nice - but I don't see the use of it, and there are no applications of the general result. (The $\delta(\sin x)$ example is my own.)
Is there any application of the result that you could share?
One classic example is to consider $f(x) = x^2 - a^2$, which yields $f'(x) = 2 \, x$ and $$\delta(x^2 - a^2) = \frac{1}{2 \, |a|} \, \left( \delta(x-a) + \delta(x+a) \right). $$
Another example is $$\delta(\cosh(ax)) = \frac{1}{|a|} \, \sum_{k=-\infty}^{\infty} \delta\left(x - \frac{(2 \, k +1) \, \pi \, i}{2 \, a} \right).$$