Question

If cot2x3+tanx3= coseckx3,cot⁡2x3+tan⁡x3= coseckx3,

• A. $2$
• B. $2-\pi$
• C. $\pi -2$
• D. None of these

Answer 1

The correct option is A $2$

$\begin{array}{c}\cot \frac{2x}{3}+\tan \frac{x}{3}=\cos ec\frac{kx}{3}\\ \frac{\cos \left(\frac{2x}{3}\right)}{\sin \left(\frac{2x}{3}\right)}+\frac{\sin \left(\frac{x}{3}\right)}{\cos \left(\frac{x}{3}\right)}\\ =\frac{\cos \left(\frac{2x}{3}\right)+2{\sin }^2\left(\frac{x}{3}\right)}{\sin \left(\frac{2x}{3}\right)}\\ =\cos ec\left(\frac{2x}{3}\right)\\ K=2\\ {\tan }^{-1}\left(\tan k\right)=2\\ Hence,optionBisthecorrectanswer.\end{array}$