I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that:
$$|c_n e^{in\theta}| = |c_n|,$$
where $n$ is an index ranging from $0$ to $\infty$, $i$ is the imaginary number, $\theta$ is an angle and $c_n$ is defined as:
$$c_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(\phi)e^{-in\phi}\:d\phi,$$
where $f(\phi)$ is a piecewise-smooth $2\pi$-periodic function. How to show that $|c_n e^{in\theta}| = |c_n|$? I will provide additional information from the book if required.
This has nothing to do with Fourier series, it is just a simple fact about the modulus of complex numbers.
Given two complex numbers $z$ and $w$ we have $|zw| = |z||w|$, so $|c_ne^{in\theta}| = |c_n||e^{in\theta}|$. Furthermore, we have $|e^{in\theta}| = 1$. This can be seen by applying Euler's rule: $$|e^{in\theta}| = |\cos(n\theta) + i\sin(n\theta)| = \sqrt{\cos^2(n\theta)+\sin^2(n\theta)} = \sqrt{1} = 1.$$