Why is $\int_{-\infty}^{\infty}e^{itx}\frac{1}{\pi(1+x^2)}dx=\sqrt{\frac{2}{\pi}}\int_0^{\infty}e^{-\frac{1}{2}\left(y^2+ \frac{t^2}{y^2}\right)}dy$?

Question

I understand that there are two ways of representing $e^{-|t|}$:

$$ e^{-|t|} = \sqrt{\frac{2}{\pi}}\int_0^{\infty}e^{-\frac{1}{2}\left(y^2+ \frac{t^2}{y^2}\right)}dy $$

and also that

$$ e^{-|t|} = \int_{-\infty}^{\infty}e^{itx}\frac{1}{\pi(1+x^2)}dx $$

However, I cannot reconcile or see why these two are necessarily equal. Is there a way to see why these two are necessarily equal without resorting to just pure mechanics? Thanks.