Question

If $cotx+\cos ecx=\surd 3$ , then the principal value of $\left(x-\frac{\pi }{6}\right)$

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{2}$
• D. $\frac{\mathrm{\pi }}{6}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{6}$

Find the value of $\left(x-\frac{\pi }{6}\right)$:

Given, $cotx+\cos ecx=\surd 3$

$\Rightarrow$ $\frac{\cos x}{\sin x}+\frac{1}{\sin x}=\surd 3$

$\Rightarrow$ $\frac{(\cos x+1)}{\sin x}=\surd 3$

$\Rightarrow$$\frac{2{\cos }^2\left({\displaystyle \frac{x}{2}}\right)}{2\sin \left({\displaystyle \frac{x}{2}}\right)\cos \left({\displaystyle \frac{x}{2}}\right)}=\surd 3$

$\Rightarrow$ $\frac{\cos \left({\displaystyle \frac{x}{2}}\right)}{\sin \left({\displaystyle \frac{x}{2}}\right)}=\surd 3$

$\Rightarrow$ $cot\left(\frac{x}{2}\right)=\surd 3$

$\Rightarrow$ $\tan \left(\frac{x}{2}\right)=\frac{1}{\surd 3}$

$\Rightarrow$ $\frac{x}{2}=n\pi +\frac{\pi }{6}$

$\Rightarrow$ $x=2n\pi +\frac{\pi }{3}$

$\Rightarrow$ $x–\frac{\pi }{6}=2n\pi +\frac{\pi }{6}$

$\therefore$The principal value of $x-\frac{\mathrm{\pi }}{6}$ for $n=0$ is $\frac{\mathbf{\pi }}{\mathbf{6}}$

Hence, Option ‘D’ is Correct.