Animated Solution for Mathematics - Definite Integration: Find the area of the region bounded by the curve $C:y=\tan x$, tangent drawn to $C$ at $x=\frac{\pi }{4}$ and the $x$-axis.

We are tasked with finding the area of a region bounded by the curve $y=\tan x$, the tangent line drawn to this curve at $x=\frac{\pi }{4}$, and the $x$-axis. This problem requires a precise combination of coordinate geometry and integral calculus.

First, we identify the point of tangency on the curve $y=\tan x$. At $x=\frac{\pi }{4}$, the function value is:

Thus, our point of tangency, $P$, is located at $\left(\frac{\pi }{4},1\right)$. This point serves as the anchor for our tangent line.

To define the tangent line, we calculate the derivative of the function $y=\tan x$, which is $\frac{dy}{dx}={\sec }^{2}x$. Evaluating this at $x=\frac{\pi }{4}$ gives the slope $m$:

Using the point-slope form $y-y_1=m(x-x_1)$, the equation of the tangent line is:

The tangent line intersects the $x$-axis at point $L$. Setting $y=0$ in the tangent equation, we solve for $x$:

By dropping a perpendicular from $P$ to the $x$-axis at $M\left(\frac{\pi }{4},0\right)$, we form a right-angled triangle $PLM$. The base $LM$ is $\frac{\pi }{4}-(\frac{\pi }{4}-\frac{1}{2})=\frac{1}{2}$, and the height is $1$.

The area of triangle $PLM$ is:

We now calculate the area under the curve $y=\tan x$ from $x=0$ to $x=\frac{\pi }{4}$ using a definite integral:

Evaluating this expression, we obtain:

The target area is the region under the curve minus the area of the triangle $PLM$. Subtracting the triangle's area from the integral result, we get:

This simplifies to the final answer: