Correct Answer - Option 2 : 4/3
Concept:
\(\rm 2tan^{-1}x = tan^{-1}\left ( \frac{2x}{1-x^{2}} \right )\)
\(\rm cot^{-1}x = tan^{-1}\frac{1}{x}\)
Calculation:
\(\rm cot\left \{ 2tan^{-1}\left ( \frac{1}{3} \right ) \right \}\)
As we know that , \(\rm 2tan^{-1}x = tan^{-1}\left ( \frac{2x}{1-x^{2}} \right )\)
= \(\rm cot\left \{ tan^{-1}\left ( \frac{2\times \frac{1}{3}}{1-(\frac{1}{3})^{2}} \right ) \right \}\)
= \(\rm cot \left \{ tan^{-1}\left ( \frac{\frac{2}{3}}{1-\frac{1}{9}} \right ) \right \}\)
= cot \(\rm \left \{ tan^{-1}\left ( \frac{3}{4} \right ) \right \}\)
As we know, \(\rm cot^{-1}x = tan^{-1}\frac{1}{x}\)
= cot \(\rm \left \{ cot^{-1}\frac{4}{3} \right \}\)
= 4/3
The correct option is 2.
