= Solution
Multiply $H$ by the exponential form of the sine and use the <modulation property of the Fourier transform>:
$$
\widetilde f(\omega)=\frac1{2i}[\widetilde H(\omega-\omega_0)-\widetilde H(\omega+\omega_0)].
$$
Inserting part (iii) gives
$$
\boxed{\widetilde f(\omega)=\frac12\left[\operatorname{PV}\frac1{\omega+\omega_0}-\operatorname{PV}\frac1{\omega-\omega_0}\right]+\frac\pi{2i}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)].}
$$
The reciprocal terms are real principal-value distributions and the delta terms are imaginary. Equivalently, exponential damping $H(t)e^{-\epsilon t}$ gives $\omega_0/[(\epsilon+i\omega)^2+\omega_0^2]$, whose distributional limit is the boxed expression. The shifted-difference formula remains valid at $\omega_0=0$, where all terms cancel.
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