Question

Find the value of x, if $5{\tan }^{-1}x+2co{t}^{-1}x=2\pi .$

• A. $1$
• B. $\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $0$

Answer 1

The correct option is B $\sqrt{3}$

$5{\tan }^{-1}x+2co{t}^{-1}x=2\pi$
$\Rightarrow 3{\tan }^{-1}x+2\left({\tan }^{-1}x+co{t}^{-1}x\right)=2\pi$
$\mathrm{We}\mathrm{know}\mathrm{that}{\tan }^{-1}x+co{t}^{-1}x=\frac{\pi }{2}\Rightarrow 3{\tan }^{-1}x+2\times \frac{\pi }{2}=2\pi \Rightarrow 3{\tan }^{-1}x=\pi \Rightarrow {\tan }^{-1}x=\frac{\pi }{3}\Rightarrow x=\tan \frac{\pi }{3}\Rightarrow x=\sqrt{3}.$