Question:
$\cot \left(\frac{\pi}{4}-2 \cot ^{-1} 3\right)=$
(a) 7
(b) 6
(c) 5
(d) none of these
Solution:
(a) 7
Let $2 \cot ^{-1} 3=y$
Then, $\cot \frac{y}{2}=3$
$\cot \left(\frac{\pi}{4}-2 \cot ^{-1} 3\right)=\cot \left(\frac{\pi}{4}-y\right)$
$=\frac{\cot \pi / 4 \cot y+1}{\cot y-\cot \pi / 4}$
$=\frac{\cot y+1}{\cot y-1}$
$=\frac{\frac{\cot ^{2} \frac{y}{2}-1}{2 \cot \frac{y}{2}}+1}{\frac{\cot ^{2} \frac{y}{2}-1}{2 \cot \frac{y}{2}}-1}$
$=\frac{\cot ^{2} \frac{y}{2}+2 \cot \frac{y}{2}-1}{\cot ^{2} \frac{y}{2}-2 \cot \frac{y}{2}-1}$
$=\frac{9+6-1}{9-6-1}$
$=7$
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