Clarification or guidance on exercise involving integral transform

Question

I have an exercise that I'm running circles around, and I'd like to state the problem, then discuss what I've attempted, and ask for some guidance. The problem is to prove the following equality: $$\int_0^\infty e^{-2at}\sin^2(at)dt =\frac{1}{\pi}\int_0^\infty \frac{a^2}{4a^4+\omega^4}d\omega $$ I assumed, perhaps incorrectly, that this would involve a Fourier transform, since the integrand on the left converges to zero as $t$ approaches infinity. Also, the change of variable to $\omega$ seems traditional for the Fourier transform. Then I wondered if Parseval's relation could be employed, since we have the equality of two integrals and particularly because the integrand on the left is the square of $e^{-at}\sin(at)$. I'm also not sure if Euler's Identity will be needed, and if so, how to resolve the imaginary part since I didn't seem to obtain an odd function in the integration (assuming that if Parseval's applies here, I'd still need to carry out a Fourier transform -- let alone that we are integrating from $0$ to $\infty$). I'm also left wondering if I'm instead supposed to compute both integrals and show that they are equal to the same result. Fourier transform exercises usually result in a function, not another integral statement. And even though we typically use the variable $s$ for Laplace transforms, I wonder if that is the more viable option here. Brute-force integration resulted in me applying Integration by Parts twice over and still yielding a trigonometric antiderivative that doesn't gel with the infinite limit of integration. On balance, I have a couple of pages of dense scratch work and no conclusions to show for it. Please offer specific advice if you would be so kind.