Question

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

Answer 1

Answer: 210

Let $ta{n}^{-1}\frac{1}{3}=\alpha$ and $ta{n}^{-1}2\sqrt{2}=\beta$
$\therefore sin\left(2ta{n}^{-1}\frac{1}{3}\right)+cos\left(ta{n}^{-1}2\sqrt{2}\right)=sin2\alpha +cos\beta$
$=\frac{2tan\alpha }{1+ta{n}^2\alpha }+\frac{1}{\sqrt{1+ta{n}^2\beta }}$
$=\frac{2\times 13}{1+119}+\frac{1}{\sqrt{1+8}}=\frac{14}{15}$
$\frac{p}{q}=\frac{14}{15}$
$\Rightarrow p=14,q=15,$
$\therefore pq=14\times 15=210$