October 2021 Paper 3 Q15

15

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Adding arctangents” are reproduced below; the line numbers are those printed on the Insert.

Line 7
It can be shown that \(\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) = \arctan 1\).

Line 29
For any positive \(x\), \(\arctan x + \arctan\left(\dfrac{1}{x}\right) = \dfrac{\pi}{2}\).

Line 37
\(\arctan x + \arctan y = \arctan\left(\dfrac{x + y}{1 - xy}\right) + \pi\), when \(xy > 1\) and \(x, y > 0\)

Lines 41–42
• \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\). This can be proved by using \(\arctan x + \arctan\left(\dfrac{1}{x}\right) = \dfrac{\pi}{2}\) together with \(\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) = \arctan 1\).

Prove that \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\), as given in line 41. [4]