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Question
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\]
Options
7
6
5
none of these
Solution
(a) 7
Let \[2 \cot^{- 1} 3 = y\]
Then,
\[\cot\frac{y}{2} = 3\]
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) = \cot\left( \frac{\pi}{4} - y \right)\]
\[ = \frac{\cot\frac{\pi}{4}\cot{y} + 1}{\cot{y} - \cot\frac{\pi}{4}}\]
\[ = \frac{\cot{y} + 1}{\cot{y} - 1} \]
\[ = \frac{\frac{\cot^2 \frac{y}{2} - 1}{2\cot\frac{y}{2}} + 1}{\frac{\cot^2 \frac{y}{2} - 1}{2\cot\frac{y}{2}} - 1}\]
\[ = \frac{\cot^2 \frac{y}{2} + 2\cot\frac{y}{2} - 1}{\cot^2 \frac{y}{2} - 2\cot\frac{y}{2} - 1}\]
\[ = \frac{9 + 6 - 1}{9 - 6 - 1}\]
\[ = 7\]
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