Demodulating upper sideband (USB) signals
Apart from scale factors, your USB signal is

$$f(t)=m(t)\cos(2\pi\nu_ct)-\hat{m}(t)\sin(2\pi\nu_ct)\tag{1}$$

(as you've correctly stated in your question). Now if you multiply with a (coherent) carrier $2\cos(2\pi\nu_ct)$, you get

$$\begin{align}g(t)&amp;=2\big[m(t)\cos(2\pi\nu_ct)-\hat{m}(t)\sin(2\pi\nu_ct)\big]\cos(2\pi\nu_ct)\\
&amp;=2m(t)\cos^2(2\pi\nu_ct)-2\hat{m}(t)\sin(2\pi\nu_ct)\cos(2\pi\nu_ct)\\&amp;=m(t) (1+\cos(4\pi\nu_ct))-\hat{m}(t)\sin(4\pi\nu_ct)\tag{2}\end{align}$$

Applying a low pass filter to $(2)$ will filter out all terms at $2\nu_c$, leaving you with the message signal $m(t)$.