Solve this
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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