Fourier Transform of the Hilbert Transform of cos(t) (using Fourier time-shifting property)
I forgot about the property of a dirac delta function: $f(t)\delta (t)=f(t)|_{t=0} \delta (t)=f(0)\delta (t)$

$Y(\omega)=F\{x(t-\frac{\pi}{2})\}=X(\omega)e^{-j\omega\frac{\pi}{2}}=(\pi \delta(\omega-1)+\pi \delta(\omega+1))e^{-j\omega\frac{\pi}{2}}=\pi \delta(\omega-1)e^{-j(1)\frac{\pi}{2}}+\pi \delta(\omega+1)e^{-j(-1)\frac{\pi}{2}}=-j\pi \delta(\omega-1)+j\pi \delta(\omega+1)$