Question
Prove that:
$\cot12^\circ\cot38^\circ\cot52^\circ\cot60^\circ\cot78^\circ=\frac{1}{\sqrt{3}}$
$\cot12^\circ\cot38^\circ\cot52^\circ\cot60^\circ\cot78^\circ=\frac{1}{\sqrt{3}}$
✓
Answer
$\text{L.H.S.}=\cot12^\circ\cot38^\circ\cot52^\circ\cot60^\circ\cot78^\circ$
$=\tan(90^\circ-12^\circ)\tan(90^\circ-38^\circ)\times\cot52^\circ\times\frac{1}{\sqrt{3}}\times\cot78^\circ$
$=\frac{1}{\sqrt{3}}\times\tan78^\circ\times\tan52^\circ\times\cot52^\circ\times\cot78^\circ$
$=\frac{1}{\sqrt{3}}\times\tan786\circ\times\tan52^\circ\times\frac{1}{\tan52^\circ}\times\frac{1}{\tan78^\circ}$
$=\frac{1}{\sqrt{3}}$
$=\text{R.H.S.}$
$=\tan(90^\circ-12^\circ)\tan(90^\circ-38^\circ)\times\cot52^\circ\times\frac{1}{\sqrt{3}}\times\cot78^\circ$
$=\frac{1}{\sqrt{3}}\times\tan78^\circ\times\tan52^\circ\times\cot52^\circ\times\cot78^\circ$
$=\frac{1}{\sqrt{3}}\times\tan786\circ\times\tan52^\circ\times\frac{1}{\tan52^\circ}\times\frac{1}{\tan78^\circ}$
$=\frac{1}{\sqrt{3}}$
$=\text{R.H.S.}$
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