Question
Evaluate:
$=\frac{\sin18^\circ}{\cos72^\circ}+\sqrt{3}\ \{\tan10^\circ\tan30^\circ\tan40^\circ\tan50^\circ\tan80^\circ\}$
$=\frac{\sin18^\circ}{\cos72^\circ}+\sqrt{3}\ \{\tan10^\circ\tan30^\circ\tan40^\circ\tan50^\circ\tan80^\circ\}$
✓
Answer
We will use the properties of complementary angles.
$=\frac{\sin18^\circ}{\cos72^\circ}+\sqrt{3}\ (\tan10\tan30\tan40\tan50\tan80)$
$ =\frac{\sin18^\circ}{\sin18^\circ}+\sqrt{3}\ (\cot80\tan30\cot50\tan50\tan80)$
$ =1+\sqrt{3}\Big(\frac{1}{\sqrt{3}}\Big)$
$ =2$
$=\frac{\sin18^\circ}{\cos72^\circ}+\sqrt{3}\ (\tan10\tan30\tan40\tan50\tan80)$
$ =\frac{\sin18^\circ}{\sin18^\circ}+\sqrt{3}\ (\cot80\tan30\cot50\tan50\tan80)$
$ =1+\sqrt{3}\Big(\frac{1}{\sqrt{3}}\Big)$
$ =2$
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