Inverses of Cosine Theorem:
For $$F_c(\omega) = \int_{0}^{\infty} f(t) \cos (\omega t) dt,$$ $$f(t)=\frac{2}{\pi} \int_{0}^{\infty} F_c(\omega)\cos (\omega t) d\omega .$$
I want proof of this theorem, but I can't find it in textbooks.
I want to seek a hint for this proof.
What I tried is:
$$F_c(\omega) = \int_{0}^{\infty} f(t)\cos (\omega t) dt$$ \begin{align}\int_{0}^{\infty} F_c (\omega) \cos (\omega t) d\omega &=\int_{0}^{\infty} \cos \omega t \left(\int_0^{\infty} f(t) \cos (\omega t) dt \right) d \omega \\&= \int_{0}^{\infty} \cos (\omega t) d \omega \int_{0}^{\infty} f(t) \cos (\omega t) dt\\ \end{align} but I don't know how to do further steps.
Edit 1: Inverses of Cosine Theorem:
For $$F_c(\omega) = \int_{0}^{\infty} f(t) \cos (\omega t) dt,$$ $$f(\tau)=\frac{2}{\pi} \int_{0}^{\infty} F_c(\omega)\cos (\omega \tau) d\omega .$$ My Approach: $$F_c(\omega) = \int_{0}^{\infty} f(t)\cos (\omega t) dt$$ \begin{align}\int_{0}^{\infty} F_c (\omega) \cos (\omega \tau) d\omega &=\int_{0}^{\infty} \cos \omega \tau \left(\int_0^{\infty} f(t) \cos (\omega t) dt \right) d \omega \\&= \int_{0}^{\infty} \cos (\omega \tau) d \omega \int_{0}^{\infty} f(t) \cos (\omega t) d\tau\\ \end{align} but I don't know how to do further steps.
Edit 2:
Textbook Used: P.P.G. Dyke, "An Introduction to Laplace Transforms and Fourier Series", Springer (P. 135)