Question
Prove that:
$\tan5^\circ\tan25^\circ\tan30^\circ\tan65^\circ\tan85^\circ=\frac{1}{\sqrt{3}}$
$\tan5^\circ\tan25^\circ\tan30^\circ\tan65^\circ\tan85^\circ=\frac{1}{\sqrt{3}}$
✓
Answer
$\text{L.H.S.}=\tan5^\circ\tan25^\circ\tan30^\circ\tan65^\circ\tan85^\circ$
$=\tan(90^\circ-85^\circ)\tan(90^\circ-65^\circ)\times\frac{1}{\sqrt{3}}\times\frac{1}{\cot65^\circ}\frac{1}{\cot85^\circ}$
$=\cot85^\circ\cot65^\circ\frac{1}{\sqrt{3}}\frac{1}{\cot65^\circ}\frac{1}{\cot85^\circ}$
$=\frac{1}{\sqrt{3}}$
$=\text{R.H.S.}$
$=\tan(90^\circ-85^\circ)\tan(90^\circ-65^\circ)\times\frac{1}{\sqrt{3}}\times\frac{1}{\cot65^\circ}\frac{1}{\cot85^\circ}$
$=\cot85^\circ\cot65^\circ\frac{1}{\sqrt{3}}\frac{1}{\cot65^\circ}\frac{1}{\cot85^\circ}$
$=\frac{1}{\sqrt{3}}$
$=\text{R.H.S.}$
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