I am trying to solve the following exercise
Use $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$ to calculate the Fourier-Transformation of $\sin x$, $\cos x$, $\sinh x$ and $\cosh x$
Now I am a little bit confused, because the fourier transformation of $\sin x$ is simply $\sin x$, of $\cos x$ is $\cos x$, of $\sinh x$ it is $-i \sin(ix)$ and of $\cosh(x)$ it is $\cos(ix)$ simply by definition, so why should I use the relation $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$?
Because there are formulae: $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}\quad\cos x=\frac{e^{ix}+e^{-ix}}2$$ $$\sinh x=\frac{e^x-e^{-x}}2\quad\cosh x=\frac{e^x+e^{-x}}2$$ Use them and the Fourier transform of $e^{bx}$