Question

The value of ${\sin }^{-1}\frac{4}{5}+2{\tan }^{-1}\frac{1}{3}=$

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. None of these

Answer 1

The correct option is A $\frac{\pi }{2}$

${\sin }^{-1}\frac{4}{5}={\tan }^{-1}\frac{4}{3},2{\tan }^{-1}\frac{1}{3}={\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{3}={\tan }^{-1}\left(\frac{2/3}{1-1/9}\right)={\tan }^{-1}\frac{3}{4}$

Hence
${\sin }^{-1}\frac{4}{5}+2{\tan }^{-1}\frac{1}{3}={\tan }^{-1}\frac{4}{3}+{\tan }^{-1}\frac{3}{4}={\tan }^{-1}\frac{4}{3}+{\cot }^{-1}\frac{4}{3}=\frac{\pi }{2}$