Correct Answer - Option 3 :
\(\rm \cot^{-1}{41\over3}\)
Concept:
- \(\rm \tan^{-1}x + \tan^{-1}y=\tan^{-1}{x+y\over 1-xy}\)
- \(\rm cot^{-1}{x} = {\pi\over2}- \tan^{-1}{x}\)
- \(\rm 2tan^{-1}\ x =tan^{-1} ({\frac {2x}{1\ -\ x^2}})\)
Calculation:
S = \(\rm cot^{-1}{1\over3} - 2 \tan^{-1}{2\over3} \)
S = \(\rm \left[{\pi\over2}-\tan^{-1}{1\over3}\right] - \tan^{-1}{{2\over3}+{2\over3}\over1-{2\over3}\times{2\over3}}\)
S = \(\rm {\pi\over2}- \left[ \tan^{-1}{12\over5}+\tan^{-1}{1\over3}\right]\)
S = \(\rm {\pi\over2}- \left[ \tan^{-1}{{12\over5}+{1\over3}\over1-{12\over5}\times{1\over3}}\right]\)
S = \(\rm {\pi\over2}- \left[ \tan^{-1}{41\over3}\right]\)
S = \(\rm \cot^{-1}{41\over3}\)
