Question

The numerical value of $\tan {20}^{\circ }\tan {80}^{\circ }\cot {50}^{\circ }$ is equal to

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

Answer 1

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

$\tan 20\tan 80\cot 50$

$=\frac{\tan (30-10)\tan (90-10)}{\tan (60-10)}=\frac{\frac{\tan 30-\tan 10}{1+\tan 30\tan 10}}{\frac{\tan 60-\tan 10}{1+\tan 60\tan 10}}cot10$

$=\frac{\frac{\frac{1}{\sqrt{3}}-\tan 10}{1+\frac{1}{\sqrt{3}}\tan 10}}{\tan 10\frac{\sqrt{3}-\tan 10}{1+\sqrt{3}\tan 10}}=\frac{\frac{1-\sqrt{3}\tan 10}{\sqrt{3}+\tan 10}}{\tan 10\cdot \frac{\sqrt{3}-\tan 10}{1+\sqrt{3}\tan 10}}$

$=\frac{1-3{\tan }^{2}10}{\tan 10(3-{\tan }^{2}10)}=\frac{1-3{\tan }^{2}10}{3\tan 10-{\tan }^{3}10}=\frac{1}{\tan 3\left(10\right)}$

$=\frac{1}{\tan 30}=\sqrt{3}$