How to calculate the Fourier coefficients

Question

How to calculate the Fourier coefficients, of a given input : $$u(t)=\sum_{k=-\infty}^{+\infty}\overline u_k e^{ik\omega_0t}$$ of this output: $$v(t)=u(t)(1-cos^2(\omega_0t))$$ So how is the mathematical definition of this, maybe started from the definition of Fourier series. So how can I start this ?

Answer 1

$$v(\Omega)=\frac{1}{2\pi}\int_{-\pi}^{\pi}u(t)e^{{i}\Omega{t}}dt-\frac{1}{2\pi}\int_{-\pi}^{\pi}u(t)\cos^{2}(\omega_{0}t)e^{{i}\Omega{t}}dt=$$ $$=\frac{1}{2}u(\Omega)-\frac{1}{2\pi}\int_{-\pi}^{\pi}u(t)\frac{e^{2i\omega_{0}t}+e^{-2i\omega_{0}t}}{4}e^{{i}\Omega{t}}dt=$$ $$=\frac{1}{2}u(\Omega)-\frac{1}{4}u(\Omega+2\omega_{0})-\frac{1}{4}u(\Omega-2\omega_{0})$$