Question

How do you evaluate `arctan(1/2)+arctan(1/3)`?

Answer 1

By using the formula for the tangent of a sum, we find that the given sum is just `\pi/4`.

Explanation:

Let `tan\theta_1=1/2, tan\theta_2=1/3`. From the formula for the tangent of a sum:`tan(\theta_1+\theta_2)={tan\theta_1+tan\theta_2}/{1-tan\theta_1tan\theta_2}`
`={1/2+1/3}/{1-(1/2)(1/3)}=1`

Since each individual term `theta_1,\theta_2` is between `0` and `\pi/4`, their sum must be between `0` and `\pi/2`. So then:

`\theta_1+\theta_2=arctan(1)=\pi/4`.