Question

$\cot \left(\frac{\mathrm{\pi }}{4}-2{\cot }^{-1}3\right)=$
(a) 7
(b) 6
(c) 5
(d) none of these

Answer 1

(a) 7

Let $2{\cot }^{-1}3=y$
Then, $\cot \frac{y}{2}=3$

$$\begin{gathered} \cot \left(\frac{\mathrm{\pi }}{4}-2{\cot }^{-1}3\right)=\cot \left(\frac{\mathrm{\pi }}{4}-y\right) \\ =\frac{\cot {\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\cot y+1}{\cot y-\cot \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} \\ =\frac{{\displaystyle \cot }{\displaystyle y}+1}{\cot y-1} \\ =\frac{{\displaystyle \frac{{\cot }^2{\displaystyle \frac{y}{2}}-1}{2\cot {\displaystyle \frac{y}{2}}}}+1}{{\displaystyle \frac{{\cot }^2{\displaystyle \frac{y}{2}}-1}{2\cot {\displaystyle \frac{y}{2}}}}-1} \\ =\frac{{\cot }^2{\displaystyle \frac{y}{2}}+2\cot {\displaystyle \frac{y}{2}}-1}{{\cot }^2{\displaystyle \frac{y}{2}}-2\cot {\displaystyle \frac{y}{2}}-1} \\ =\frac{9+6-1}{9-6-1} \\ =7 \end{gathered}$$