Trigonometry : Fun Proof

1. Find the value [without using any calculator] of :
arctan(1) + arctan(2) + arctan(3)
You will get a surprising value.

2. What is again : [Proof without a calculator]
arctan(1/2) + arctan(1/4) ??

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I want the shortest possible proof. I really got surprised after I found its value.
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Is there a logical explanation for such an answer. I am talking about logical proof besides mathematical proof.
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1.
arctan(1) = π/4
arctan(2) + arctan(3) = π + arctan [(2 + 3) / (1 - 2*3)] = π - π/4
=> arctan(1) + arctan(2) + arctan(3) = π.

2.
arctan(1/2) + arctan(1/4)
= arctan [(1/2 + 1/4)/(1 - (1/2)*(1/3))]
= arctan(1)
= π/4.

gôhpihán pointed out the typing error in problem 2) and hence incorrect answer. It should be
arctan [(1/2 + 1/4)/(1 - (1/2)*(1/4))]
= arctan(6/7)
Surprised at 4 TUs for incorrect answer!!

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It seems that the asker wanted to ask
arctan(1/2) + arctan(1/3) which can be proved to be π/4.

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Yes Yes π/4

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