Value of \( \cot\left\{\displaystyle\frac{\pi}{4}-2\cot^{-1}3\right\} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 7
Let \( \theta=\cot^{-1}3 \), so \( \tan\theta=\frac{1}{3} \). By the double-angle formula,
\( \displaystyle \tan2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta} \)
\( \displaystyle =\frac{\frac{2}{3}}{\frac{8}{9}}=\frac{3}{4} \)
By the subtraction formula for tan,
\( \displaystyle \tan\left(\frac{\pi}{4}-2\theta\right)=\frac{1-\tan2\theta}{1+\tan2\theta} \)
\( \displaystyle =\frac{\frac{1}{4}}{\frac{7}{4}}=\frac{1}{7} \)
So \( \cot\left(\frac{\pi}{4}-2\theta\right)=7 \).