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Question

The value of $\cot (\frac{π}{4} - 2 \cot^{- 1} 3),$ is

Moderate

Solution

$\cot (\frac{π}{4} - 2 \cot^{- 1} 3)$
$= \frac{1}{\sin (\frac{π}{4} - 2 \tan^{- 1} \frac{1}{3})} = \frac{1 + \tan (2 \tan^{- 1} \frac{1}{3})}{1 - \tan (2 \tan^{- 1} \frac{1}{3})}$
$= \frac{1 - \tan (\tan^{- 1} \frac{3}{4})}{1 - \tan (\tan^{- 1} \frac{3}{4})} = 7 [∵ 2 \tan^{- 1} \frac{1}{3} = \tan^{- 1} (\frac{2 / 3}{1 - \frac{1}{9}}) = \tan^{- 1} \frac{3}{4}]$

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