Question

Evaluate: $\frac{sin{18}^0}{cos{72}^0}+\sqrt{3\left\{tan{10}^{0}tan{30}^{0}tan{40}^{0}tan{50}^{0}tan{80}^0\right\}}$

Answer 1

$\frac{\sin 18}{\cos 72}+\sqrt{3\left(\tan 10\tan 30\tan 40\tan 50\tan 80\right)}$

$\frac{\sin 18}{\cos (90-18)}+\sqrt{3(\tan 10\tan 30+\tan 40\cot 40\cot 10)}$

$1+\sqrt{3\tan 30}$

$1+\sqrt{3\times \frac{1}{\sqrt{3}}}$

$=1+3^{\frac{1}{4}}$.