Question

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

Answer 1

`tan x = 1/2 --> x = 26.57` deg

`tan y = 1/3 --> y = 18.43` deg

`(x + y) = 45 deg = pi/4`

Answer 2

Let `A = arctan(1/2)`, so that `-pi/2 < A < pi/4` and `tan A = 1/2`

(Note that, since `tan A` is positive, we can further conclude that `< A < pi/4` )

Let `B = arctan(1/3)`, so that `0 < B < pi/4` and `tan B = 1/3`

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

We need to show that `A+B = pi/4`.

Use the formula for `tan(A+B)` to show that `tan (A+B)= 1`.

Conclude that `A+B = pi/4`.