Find the equation of the tangent line $g$ to the graph of the function $y = \tan x$ at the point $(\frac{\pi}{4}, 1)$.

1. Problem Description

Find the equation of the tangent line $g$ to the graph of the function $y = \tan x$ at the point $(\frac{\pi}{4}, 1)$.

2. Solution Steps

First, we need to find the derivative of the function $y = \tan x$. The derivative is:

$\frac{dy}{dx} = \sec^2 x$.

Next, we evaluate the derivative at the given point $x = \frac{\pi}{4}$. This gives us the slope of the tangent line at that point.

$m = \sec^2 (\frac{\pi}{4})$.

Since $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, we have $\sec(\frac{\pi}{4}) = \frac{2}{\sqrt{2}} = \sqrt{2}$.

Therefore, $m = (\sqrt{2})^2 = 2$.

The equation of the tangent line is given by the point-slope form:

$y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point $(\frac{\pi}{4}, 1)$ and $m$ is the slope.

$y - 1 = 2(x - \frac{\pi}{4})$

$y - 1 = 2x - \frac{\pi}{2}$

$y = 2x - \frac{\pi}{2} + 1$

$y = 2x + (1 - \frac{\pi}{2})$

3. Final Answer

The equation of the tangent line is $y = 2x + (1 - \frac{\pi}{2})$.