I am trying to find the radon transform of the gaussian function $$f(x,y) = e^{-(x^2 + y^2)}$$
Now, I am using the formula for radon transform as $$ [\mathcal{R}f]{(\rho, \theta)} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \delta(x \cos(\theta) + y \sin(\theta) - \rho) dx dy $$
When I am solving it, I am replacing $y = (\rho - x \cos \theta)/\sin \theta$ and this boils down the main integral to a single gaussian integral but my final answer results to what is expected but with a $|\sin \theta|$ multiplied with it.
How do i prove analytically that it is equal to $\sqrt{\pi} e^{-\rho^2}$? Can someone help me with the calculations?