I am asked to simplify the complex expression $$\frac{1}{2}(|{e^{i{\theta}}-1}^2|+|{e^{i{\theta}}+1}|)$$
I have gotten to $$\frac{1}{2}((2-2cos\theta)+(2+2cos\theta))$$ 1. Do I expand to get $$\frac{1}{2}(4)$$ OR 2. Do I factor out the 2 and get $$\frac{1}{2}(2(1-cos\theta)+2(1+cos\theta))$$
Is this answer complete? Additional question: How would the method change if the argument was negative? i.e. if it was $$\frac{1}{2}(|{e^{-i{\theta}}-1}^2|+|{e^{-i{\theta}}+1}|)$$
The complet answer is $2$. That follows from both of your approaches. That is$$(\forall\theta\in\mathbb R):\lvert e^{i\theta}-1\rvert^2+\lvert e^{-i\theta}-1\rvert^2=2.$$Note that this is for all real numbers $\theta$. Therefore, it makes no sense to ask what happens if we deal with $-\theta$ instead of $\theta$.