Question

If $\tan {40}^{\circ }+2\tan {10}^{\circ }=\cot x$, where $x\in (0,\pi /2)$, then the possible value of $x$ is

• A. ${75}^{\circ }$
• B. ${85}^{\circ }$
• C. ${30}^{\circ }$
• D. ${40}^{\circ }$

Answer 1

The correct option is D ${40}^{\circ }$

We know that,
$\tan {50}^{\circ }=\tan ({40}^{\circ }+{10}^{\circ })\Rightarrow \tan {50}^{\circ }=\frac{\tan {40}^{\circ }+\tan {10}^{\circ }}{1-\tan {40}^{\circ }\tan {10}^{\circ }}\Rightarrow \tan {50}^{\circ }-\tan {50}^{\circ }\tan {40}^{\circ }\tan {10}^{\circ }=\tan {40}^{\circ }+\tan {10}^{\circ }\Rightarrow \cot {40}^{\circ }-\tan {10}^{\circ }=\tan {40}^{\circ }+\tan {10}^{\circ }(\because \tan {50}^{\circ }=\cot {40}^{\circ })\Rightarrow \cot {40}^{\circ }=\tan {40}^{\circ }+2\tan {10}^{\circ }\Rightarrow \cot {40}^{\circ }=\cot x\Rightarrow x={40}^{\circ }$