Question

The value of $2{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{2}=$

• A. ${\cot }^{-1}2$
• B. ${60}^{\circ }$
• C. ${30}^{\circ }$
• D. ${\tan }^{-1}2$

Answer 1

The correct option is D ${\tan }^{-1}2$

$2{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{2}={\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{2}={\tan }^{-1}\left(\frac{\frac{2}{3}}{1-\frac{1}{9}}\right)+{\tan }^{-1}\left(\frac{1}{2}\right)$

$={\tan }^{-1}\left(\frac{3}{4}\right)+{\tan }^{-1}\left(\frac{1}{2}\right)={\tan }^{-1}\left(\frac{\frac{1}{2}+\frac{3}{4}}{1-\frac{1}{2}\times \frac{3}{4}}\right)={\tan }^{-1}\left(2\right)$