$|i-(\cos(\theta)+i\sin(\theta)|^2=4\sin^2(\frac{\pi}{4}-\frac{\theta}{2})$

Question

In the textbook Fuchsian Groups the author calculated what I believe to be $|i-(\cos(\theta)+i\sin(\theta)|^2=4\sin^2(\frac{\pi}{4}-\frac{\theta}{2})$ then reduced it to $2-2\sin(\theta)$. I calculated using the sum of squares formula $\cos^2(\theta)+(1-\sin(\theta))^2=2-2\sin(\theta)$ by Pythagorean identity. But more important to me is how the author thought it was more direct to start from where they did, I mean what is the identity $1-\sin(\theta)=2\sin^2(\frac{\pi}{4}-\frac{\theta}{2})$ called and why is that identity more natural then the sum of squares when finding the magnitude of the difference of two complex numbers on the unit circle?

Answer 1

Use $|i-\exp(i\theta) |=|1-\exp(i\phi)|=|-2i\sin(\phi/2)|$ with $\phi:=\theta-\pi/2$. Each $=$ uses a phase shift.

Answer 2

Let $-\cos\theta=r\cos t$

$r\sin t=1-\sin\theta$ where $r\ge0$ and $t$ is also real

Now $|r(\cos t+i\sin t)|=r|\cos t+i\sin t|=?$

Squaring and adding we get

$$r^2=(-\cos\theta)^2+(1-\sin\theta)^2=2(1-\sin\theta)$$

Now $\sin\theta=\cos(\pi/2-\theta)$

Use $\cos2x=1-2\sin^2x$