We know that we have the following equation for wave.
$$g(t)=A\cos(\omega t+\theta_0)=A\cos(2\pi ft+\theta_0)$$
The equation of frequency with respect to time will be:
$$f(t)=\frac{Bt}{\tau}+f_c-\frac{B}{2}$$
Then:
\begin{align} g(t)&=A\cos(2\pi (\frac{Bt}{\tau}+f_c-\frac{B}{2})t+\theta_0)\\ &=A\cos(2\pi(f_c-B/2)t+\mathbf{2\pi}(B/\tau)t^2+\theta_0) \end{align}
Then why that bold 2 is omitted in this answer? $$f(t)=A\cos(\theta(t))=A\cos(2\pi(f_c-B/2)t+\pi(B/\tau)t^2+\theta_0)$$