# How do you find the exact value of arctan(2) + arctan(3)?

May 25, 2015

In this way:

$\arctan 2 + \arctan 3 = \alpha$.

Let's take now the tangent of both members:

$\tan \left(\arctan 2 + \arctan 3\right) = \tan \alpha \Rightarrow$

$\frac{\tan \arctan 2 + \tan \arctan 3}{1 - \tan \arctan 2 \tan \arctan 3} = \tan \alpha \Rightarrow$

$\frac{2 + 3}{1 - 2 \cdot 3} = \tan \alpha \Rightarrow \frac{5}{1 - 6} = \tan \alpha \Rightarrow$

$\tan \alpha = - 1 \Rightarrow \alpha = \frac{3}{4} \pi + k \pi$.