Question : The value of $\cot 13° \cot 27° \cot 60° \cot 63° \cot 77°$ is:
Option 1: $\frac{1}{\sqrt{3}}$
Option 2: $0$
Option 3: $\sqrt{3}$
Option 4: $1$
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Correct Answer: $\frac{1}{\sqrt{3}}$
Solution :
Consider, $\cot 13° \cot 27° \cot 60° \cot 63° \cot 77°$
We know, $\cot(90°-\theta)=\tan\theta$ and $\tan\theta=\frac{1}{\cot\theta}$
$=\cot13°\cot27°\cot60°\cot(90°-27°)\cot(90°-13°)$
$=\cot13°\cot27°\cot60°\tan27°\tan13°$
$=\cot13°\cot27°\cot60°\cdot\frac{1}{\cot27°}\cdot\frac{1}{\cot13°}$
$=\cot60°$
$ =\frac{1}{\sqrt3}$
Hence, the correct answer is $\frac{1}{\sqrt3}$.
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