Question

The value of $sin\left(2ta{n}^{-1}\right(1/3\left)\right)+cos\left(ta{n}^{-1}2\sqrt{2}\right)$ is

• A. 12/13
• B. 13/14
• C. 14/15
• D. none of these

Answer 1

The correct option is C 14/15

Let ${\tan }^{-1}\frac{1}{3}=\alpha \Rightarrow \tan \alpha =\frac{1}{3}$
And ${\tan }^{-1}\left(2\sqrt{2}\right)=\beta \Rightarrow \tan \beta =2\sqrt{2}$
$\therefore \sin \left(2{\tan }^{-1}\frac{1}{3}\right)+\cos \left(ta{n}^{-1}\left(2\sqrt{2}\right)\right)=\sin 2\alpha +\cos \beta$

$=\frac{2\tan \alpha }{1+{\tan }^2\alpha }+\frac{1}{\sqrt{1+{\tan }^2\beta }}=\frac{2\left(\frac{1}{3}\right)}{1+\left(\frac{1}{9}\right)}+\frac{1}{\sqrt{1+8}}$

$=\frac{2}{3}\times \frac{9}{10}+\frac{1}{3}=\frac{14}{15}$