Question

If $ta{n}^{-1}x+2co{t}^{-1}x=\frac{2\pi }{3}$, then $x$ is equal to

• A. $\frac{\sqrt{3}-1}{\sqrt{3}+1}$
• B. $3$
• C. $\sqrt{3}$
• D. $\sqrt{2}$

Answer 1

Answer: (C)

$\sqrt{3}$

$ta{n}^{-1}\left(x\right)+co{t}^{-1}\left(x\right)+co{t}^{-1}\left(x\right)=\frac{2\pi }{3}$
$\Rightarrow \frac{\pi }{2}+co{t}^{-1}\left(x\right)=\frac{2\pi }{3}$
$\Rightarrow co{t}^{-1}\left(x\right)=\frac{2\pi }{3}-\frac{\pi }{2}$
$\Rightarrow co{t}^{-1}\left(x\right)=\frac{\pi }{6}$
$\Rightarrow x=cot\left(\frac{\pi }{6}\right)$
$=\sqrt{3}$