Question

Find the value of the expression
$cot{12}^{\circ }\times cot{38}^{\circ }\times cot{52}^{\circ }\times cot{60}^{\circ }\times cot{78}^{\circ }$.

• A. $\frac{2}{\sqrt{3}}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{\sqrt{3}}$
• D. $1$

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}$

Given: $cot{12}^{\circ }\times cot{38}^{\circ }\times cot{52}^{\circ }\times cot{60}^{\circ }\times cot{78}^{\circ }$

On rearranging the given equation we get,
$=(cot{12}^{\circ }\times cot{78}^{\circ })\times (cot{38}^{\circ }\times cot{52}^{\circ })\times cot{60}^{\circ }$

$=\{cot{12}^{\circ }\times cot({90}^{\circ }-{12}^{\circ }\left)\right\}\{cot{38}^{\circ }\times cot({90}^{\circ }-{38}^{\circ }\left)\right\}\times cot{60}^{\circ }$

We know that, $cot({90}^{\circ }-\theta )=tan\theta$.
Hence, $cot({90}^{\circ }-{12}^{\circ })=tan{12}^{\circ }$,
$cot({90}^{\circ }-{38}^{\circ })=tan{38}^{\circ }$ and $cot{60}^{\circ }=\frac{1}{\sqrt{3}}$
On substituting these values we get,

$=(cot{12}^{\circ }\times tan{12}^{\circ })\times (cot{38}^{\circ }\times tan{38}^{\circ })\times \frac{1}{\sqrt{3}}$

$=1\times 1\times \frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}.....[\because cot\theta \times tan\theta =1]$ .
The value of the expression
$cot{12}^{\circ }\times cot{38}^{\circ }\times cot{52}^{\circ }\times cot{60}^{\circ }\times cot{78}^{\circ }=\frac{1}{\sqrt{3}}$.