Proof using de Moivre's Theorem

Question

Let $z=\cos\theta + i\sin\theta$

Show that $$1+z = 2\cos\frac{\theta}{2}(\cos\frac{\theta}{2}+i\sin\frac{\theta}{2})$$

I don't even know how to start on this proof.

Answer 1

These are just the double angle formulas in the real and complex parts. $$\cos 2 \theta=2\cos^2\theta -1$$ $$\sin 2\theta=2\cos \theta \sin\theta$$