complex numbers - Prove the following equality

So I have the following equality involving complex numbers:
$$\frac {\sqrt 3 -1}{1-i}(1+\sqrt 3 \,i)(\cos \alpha -i\,\sin\alpha)=2\sqrt {2-\sqrt 3}\left(\cos\left(\frac {7\pi}{12}-\alpha\right)+ \sin\left(\frac {7\pi}{12}-\alpha\right)\right)$$

Guess I have to find that $\alpha$ that is missing, I thought considering both sides of the equation as single complex numbers in the trigonometric form but it seemed only to get more difficult.
I would really appreciate you help and hints, thanks.