Equation of wave in Linear frequency modulation?
If you have a signal

$$g(t)=\cos(2\pi \hat{f}(t)t)\tag{1}$$

then the function $\hat{f}(t)$ is not the instantaneous frequency of $g(t)$ (unless $\hat{f}(t)$ is constant).

If you want an instantaneous frequency $f(t)$, then the equation

$$\frac{\phi'(t)}{2\pi}=f(t)\tag{2}$$

must be satisfied, where $\phi(t)$ is the phase of the signal $g(t)$. So in order to obtain the phase $\phi(t)$, you have to integrate the desired instantaneous frequency $f(t)$. For

$$f(t)=\frac{Bt}{\tau}+f_c-\frac{B}{2}\tag{3}$$

you get

$$\frac{\phi(t)}{2\pi}=\frac{B}{2\tau}t^2+(f_c-\frac{B}{2})t+\frac{\theta_0}{2\pi}\tag{4}$$

So the signal with the desired instantaneous frequency is

$$g(t)=\cos(\phi(t))=\cos\left[\pi(Bt^2/\tau+(2f_c-B)t)+\theta_0\right]\tag{5}$$

Also take a look at this related question and its answer.