Question

The value of $sin \left(2\,tan^{-1}\frac{1}{3}\right)+cos \left(tan^{-1}2\sqrt{2}\right) is \frac{p}{q}$, then $pq$ is

Answer 1

Let $tan^{-1} \frac{1}{3} = \alpha$ and $tan^{-1} 2\sqrt{2}=\beta$
$\therefore sin \left(2\,tan^{-1}\frac{1}{3}\right)+cos\left(tan^{-1}2\sqrt{2}\right)=sin\,2\alpha+cos\,\beta$
$=\frac{2\,tan \,\alpha}{1+tan^{2} \,\alpha}+\frac{1}{\sqrt{1+tan^{2} \,\beta}} $
$=\frac{2 \times 1 3}{1+ 1 19}+\frac{1}{\sqrt{1+8}}=\frac{14}{15}$
$\frac{p}{q}=\frac{14}{15} $
$\Rightarrow p=14, q=15, $
$\therefore pq =14\times15=210$