Answer
$y=8(x-6)+\frac{\pi}{3}$
Work Step by Step
The equation of the tangent line at $(6,\frac{\pi}{3})$ is:
$$y=\frac{dy}{dx}|_{x=6}(x-6)+y(6)$$
$$y=\frac{dy}{dx}|_{x=6}(x-6)+\frac{\pi}{3}$$
Using the given equation it follows:
$$(\cos(y)+1)\frac{dy}{dx}=2t$$
$$(\cos(\frac{\pi}{3})+1)\frac{dy}{dx}|_{x=6}=2\cdot 6$$
$$(\frac{1}{2}+1)\frac{dy}{dx}|_{x=6}=2\cdot 6$$
$$\frac{dy}{dx}|_{x=6}=8$$
so:
$$y=8(x-6)+\frac{\pi}{3}$$
