# How do you show that arctan(1/2)+arctan(1/3)=pi/4?

Jun 5, 2015

$\tan x = \frac{1}{2} - \to x = 26.57$ deg

$\tan y = \frac{1}{3} - \to y = 18.43$ deg

$\left(x + y\right) = 45 \mathrm{de} g = \frac{\pi}{4}$

Jun 5, 2015

Let $A = \arctan \left(\frac{1}{2}\right)$, so that $- \frac{\pi}{2} < A < \frac{\pi}{4}$ and $\tan A = \frac{1}{2}$

(Note that, since $\tan A$ is positive, we can further conclude that $< A < \frac{\pi}{4}$ )

Let $B = \arctan \left(\frac{1}{3}\right)$, so that $0 < B < \frac{\pi}{4}$ and $\tan B = \frac{1}{3}$

(As above, since the tangent of $B$ is positive.)

We need to show that $A + B = \frac{\pi}{4}$.

Use the formula for $\tan \left(A + B\right)$ to show that $\tan \left(A + B\right) = 1$.

Conclude that $A + B = \frac{\pi}{4}$.