91Ó°ÊÓ

We are given two equations $y=\tan x$and $y=1$.

We need to find any four points of intersection of the two graphs.

We will graph the two equations and then find the first point of intersection and then using the properties of tangent function we will find the other three points.

The graph of the two equations is given as

It can be seen that the line and the curve intersect each other at infinitely many points and the first point of intersection is $\left(\frac{\mathrm{Ï€}}{4},1\right)$.

The tangent function is a cyclic function. So

$$\begin{gathered} \tan x=\tan (\mathrm{Ï€}+\mathrm{x}) \\ \mathrm{tanx}=\tan (2\mathrm{Ï€}+\mathrm{x}) \\ \mathrm{tanx}=\tan (3\mathrm{Ï€}+\mathrm{x}) \end{gathered}$$

So the x coordinates of the other points are

role="math" localid="1646463706655" $\mathrm{Ï€}+\frac{\mathrm{Ï€}}{4}=\frac{5\mathrm{Ï€}}{4}$or

$2\mathrm{Ï€}+\frac{\mathrm{Ï€}}{4}=\frac{9\mathrm{Ï€}}{4}$or

$3\mathrm{Ï€}+\frac{\mathrm{Ï€}}{4}=\frac{13\mathrm{Ï€}}{4}$

Thus the four intersecting points are $\left(\frac{\mathrm{Ï€}}{4},1\right),\left(\frac{5\mathrm{Ï€}}{4},1\right),\left(\frac{9\mathrm{Ï€}}{4},1\right),\left(\frac{13\mathrm{Ï€}}{4},1\right)$.