I have:
$$ \text{P}(x) = \frac{1}{25}|1+ 4e^{-ix}|^2 $$
I want to write this using cosine functions. According to the answer sheet, it can be rewritten in the form:
$$ \text{P}(x) = \frac{1}{25}(17 + 8\cos{x}) $$
I just can not seem to reach this result. Any help would be appreciated!
Hint: Expand it out using the definition of the magnitude of the complex number, and you will get $$\frac{1}{25} (1 + 4e^{-ix})(1 + 4e^{-ix})^*$$ where the asterisk denotes the complex conjugate. Since the complex conjugate of $e^{i\alpha}$ is just $e^{-i \alpha}$, the expression becomes $$\frac{1}{25} (1 + 4e^{-ix})(1 + 4e^{ix}) = \frac{1}{25}\left(1 + 16 + 4 [e^{ix} + e^{-ix}]\right)$$ From here, use the identity $$\cos x = \frac{e^{ix} + e^{-i x}}{2}$$ (This identity is not too difficult to prove, just use the fact that $e^{ix} = \cos x + i \sin x$. Note that $\sin$ and $\cos$ are odd and even, respectively.)