Explanation:

The given equation is:
tanx = x

Graphical Representation:

We can represent the given equation graphically, by plotting the graphs of y = tanx and y = x on the same coordinate plane. The point(s) of intersection of these two graphs will be the solution(s) of the given equation.

Quadrants:

We know that the trigonometric functions have different signs in different quadrants. Therefore, we can determine the quadrant(s) in which the solution(s) lie(s) by analyzing the signs of the trigonometric functions in each quadrant.

Analysis:

Let us analyze the signs of the trigonometric functions in each quadrant.

Quadrant I:
In this quadrant, all trigonometric functions are positive.

Quadrant II:
In this quadrant, sinx and cosecx are positive, while cosx, tanx, cotx, and secx are negative.

Quadrant III:
In this quadrant, tanx and cotx are positive, while sinx, cosx, secx, and cosecx are negative.

Quadrant IV:
In this quadrant, cosx and secx are positive, while sinx, tanx, cosecx, and cotx are negative.

Conclusion:

From the above analysis, we can see that the smallest positive root of the equation tanx = x lies in the first quadrant. This can also be verified by observing the graphs of y = tanx and y = x, which intersect in the first quadrant.