I want to find the PDF of $Y = \sin^2(X)$, where $X ∼ \text{Exponential}(\lambda)$ is a random variable. I thought i had solved this correctly by simply finding the inverse $h(Y) = X$ and taking the derivative via the chain rule. However, I was told that the solution of this problem would involve an infinite sum, but my solution did not, so I assume it was wrong. I can see why there would be an infinite sum due to the adding up of all the sine intervals, but I could not seem to figure out how to find an expression for each of these intervals that I can then sum up.
Any help would be greatly appreciated, thank you very much in advance!