Answer
$$ y=2 x+1-\frac{ \pi}{2} .$$
Work Step by Step
Since $ y=\tan x $, then $ y'=\sec^2 x $ and hence the slope at $ x=\frac{\pi}{4}$ is $ m=\sec^2 \frac{\pi}{4}=2.$ Now, the tangent line equation is given by
$$ y=2 x+c.$$
Since the curve and line coincide at $ x=\frac{\pi}{4}$, then
$$\tan \frac{\pi}{4}=2\frac{\pi}{4}+c \Longrightarrow c= 1-\frac{ \pi}{2} .$$
Hence the equation of the tangent line is given by
$$ y=2 x+1-\frac{ \pi}{2} .$$
