Show that for this function the stated is true. 
For the function
  $$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$
  show that
  $$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$

Hey everyone, I'm very new to this kind of maths and would really appreciate any help. Hopefully I can get an idea from this and apply it to other similar questions. Thank you.

Use the definition for the complex sine:
$$ \sin(z)=\frac{ e^{iz}-e^{-iz} } {2i} $$
Thus,
$$-\sqrt{2}ie^{i\frac{w}{2}}\sin\frac{w}{2} =-\sqrt{2}ie^{i\frac{w}{2}}(\frac{1}{2i}(e^{i\frac{w}{2}} - e^{-i\frac{w}{2}})) $$
Now simplify to get your result.
