MCQ
$\tan5^\circ\tan25^\circ\tan30^\circ\tan65^\circ\tan85^\circ=?$
- A$\sqrt{3}$
- ✓$\frac{1}{\sqrt{3}}$
- C$1$
- DNone of these.
✓
Answer
Correct option: B.
$\frac{1}{\sqrt{3}}$
$\tan5^\circ\tan25^\circ\tan30^\circ\tan65^\circ\tan85^\circ$
$=\tan5^\circ\times\tan25^\circ\times\frac{1}{\sqrt{3}}\times\tan(90^\circ-25^\circ)\times\tan(90^\circ-5^\circ)$
$=\tan5^\circ\times\tan25^\circ\times\frac{1}{\sqrt{3}}\times\cot25^\circ\times\cot5^\circ$
$=\tan5^\circ\cot5^\circ\tan25^\circ\cot25^\circ\times\frac{1}{\sqrt{3}}$
$=1\times1\times\frac{1}{\sqrt{3}}$
$=\frac{1}{\sqrt{3}}$
$=\tan5^\circ\times\tan25^\circ\times\frac{1}{\sqrt{3}}\times\tan(90^\circ-25^\circ)\times\tan(90^\circ-5^\circ)$
$=\tan5^\circ\times\tan25^\circ\times\frac{1}{\sqrt{3}}\times\cot25^\circ\times\cot5^\circ$
$=\tan5^\circ\cot5^\circ\tan25^\circ\cot25^\circ\times\frac{1}{\sqrt{3}}$
$=1\times1\times\frac{1}{\sqrt{3}}$
$=\frac{1}{\sqrt{3}}$
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