Finding the modulus of a complex number $1-e^{itheta }$

I have an answer for this question, but I'm not very confident with it:

$\left | 1-e^{i\theta } \right |^{2}$ i.e. find the square of the modulus of 1-$e^{i\theta }$

$e^{i\theta } = R(cos\theta +isin\theta )$, R=1

= $cos\theta +isin\theta$

In cartesian form:

$a = Rcos\theta \Rightarrow cos\theta$

$b = Rsin\theta \Rightarrow sin\theta$

$\therefore 1-e^{i\Theta } = 1-cos\theta-isin\theta$

$\Rightarrow \left | 1-e^i\theta \right | = \left | 1-cos\theta-isin\theta \right |$

Where the real part, a, = $1-cos\theta$
and the imaginary, b, = $-sin\theta$

The modulus of a complex number being $\sqrt{a^{2} +b^{2}}$

$\therefore \left | 1-cos\theta-isin\theta \right | = \sqrt{(1-cos\theta)^{2} + (-sin\theta)^{2}}$

$\Rightarrow \left | 1-cos\theta-isin\theta \right |^{2} = (1-cos\theta)^{2} + (-sin\theta)^{2} \Rightarrow 1-2cos\theta +cos^{2}\theta +sin^{2}\theta \Rightarrow (\because cos^{2}\theta +sin^{2}\theta = 1) = 2-2cos\theta$

Is this correct? Or is my approach incorrect? Thanks in advance for any feedback/help :)

Answer

Your argument is lengthy, but sound.

If just the square of the modulus is needed, you can consider that $|z|^2=z\bar{z}$, so
$$|1-e^{i\theta}|^2=(1-e^{i\theta})(1-e^{-i\theta})= 1-(e^{i\theta}+e^{-i\theta})+1=2-2\cos\theta$$

If also the argument is needed, the trick with $1-e^{i\theta}$ (but also $1+e^{i\theta}$) is to set

$$\theta=2\alpha$$
and rewrite
$$1-e^{i\theta}=1-e^{2i\alpha}= -e^{i\alpha}(e^{i\alpha}-e^{-i\alpha})=-2i(\sin\alpha)e^{i\alpha}$$
Since $0\le\theta<2\pi$, we have $0\le\alpha<\pi$, so $\sin\alpha\ge0$. Hence the modulus is $2\sin\alpha$ and, from $-i=e^{3i\pi/2}$, the argument is
$$\alpha+\frac{3\pi}{2}$$
(up to an integer multiple of $2\pi$).

Thus the square of the modulus is
$$4\sin^2\alpha=4\sin^2\frac{\theta}{2}=4\frac{1-\cos\theta}{2}= 2(1-\cos\theta)$$