Answer
Explanation given below.
Work Step by Step
Observe the three angles in the bottom right of the diagram.
The smallest right triangle has legs 1 and 1, so
$\displaystyle \tan\alpha_{1}=\frac{1}{1}$,
$\alpha_{1}=\tan^{-1}1$
The next right triangle has legs 1 and 2, so the tangent of the next angle is $\displaystyle \frac{2}{1}=2.$
$\alpha_{2}=\tan^{-1}2$
The next triangle is a right triangle, as it is constructed so that
the slope of the larger leg makes it perpendicular to the smaller leg, making it a right triangle.
So,
$\displaystyle \tan(\alpha_{3})=\frac{3\times\text{shorter leg}}{\text{shorter leg}}=3,$
that is, $\alpha_{3}=\tan^{-1}3$
The three angles add up to $180^{o}$, or in radians, $\pi$ rad,
as the terminal sides of $\alpha_{1}$ and $\alpha_{3}$ lie on the same line.
Thus,
$\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$
