Application of the time-shifting property in case of Fourier-Transform of cosine Time-shifting property: $x[n-n_d] \xrightarrow{\mathscr{F}} e^{-j\omega n_d} X(e^{j\omega}) $
Fourier-Transform of cosine-signal: $\cos(\omega_0n) \xrightarrow{\mathscr{F}} \frac{1}{2}(\delta(\omega - \omega_0) + \delta(\omega+\omega_0)) $


Combining 1. &amp; 2. together, I am getting:
$\cos(\omega_0n - \frac{\pi}{2}) \xrightarrow{\mathscr{F}} e^{-j\frac{\pi}{2}}\frac{1}{2}(\delta(\omega - \omega_0) + \delta(\omega+\omega_0)) $, but instead the Fourier-Transform of

$$\cos(\omega_0n - \tfrac{\pi}{2})$$

is 

$$\cos(\omega_0n - \tfrac{\pi}{2}) \xrightarrow{\mathscr{F}} \tfrac{1}{2}\delta(\omega - \omega_0)e^{-j\frac{\pi}{2}} + \tfrac{1}{2}\delta(\omega+\omega_0)e^{j\frac{\pi}{2}}$$

Can anyone tell what I'm doing wrong here?
Two things


Time shift equation is wrong. Should be $e^{-j \omega n_d}$. 
You need to express the cosine as a time shift $cos(\omega (n - \pi /2 /\omega))$ Time shift is not the same as phase shift and the  time shift is actually pi/2 divided by the frequency