$$
|2e^{it}-1|^2$$
I don't understand how to work this out, I know if I had for example $|2ti-1|^2$ then I would square the real and imaginary parts and add them to get the modulus squared, but here I have $|2e^{it}-1|^2$ and I don't understand what to do.
Any help would be much appreciated.
Answer
Considering that $t$ is real, we use Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$
$$|2e^{it}-1|^2=|2\cos t+2i\sin t-1|^2=(2\cos t-1)^2+(2\sin t)^2$$
This evaluates as $(5-4\cos t)$ and as you can see, the value is dependent on $t$ which is the argument of the complex number $e^{it}$ in polar form.
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