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

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Answer 1 · 2015-06-05T15:25:56

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

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

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

Answer 2 · 2015-06-05T16:16:52

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 $A+B = pi/4$ .

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