Prove that: 5^ 25^ 30^ 65^ 85^ =1 3 - Vidyadip

Question

Prove that:
$\tan5^\circ\tan25^\circ\tan30^\circ\tan65^\circ\tan85^\circ=\frac{1}{\sqrt{3}}$

✓

Answer

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

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