Question

The value of $\cos {12}^{\circ }\cos {24}^{\circ }\cos {36}^{\circ }\cos {48}^{\circ }\cos {60}^{\circ }\cos {72}^{\circ }\cos {84}^{\circ }$ is

• A. $\frac{1}{64}$
• B. $\frac{1}{512}$
• C. $\frac{1}{256}$
• D. $\frac{1}{128}$

Answer 1

The correct option is D $\frac{1}{128}$

$\cos {12}^{\circ }\cos {24}^{\circ }\cos {36}^{\circ }\cos {48}^{\circ }\cos {60}^{\circ }\cos {72}^{\circ }\cos {84}^{\circ }=\left(\cos {12}^{\circ }\cos {72}^{\circ }\cos {48}^{\circ }\right)\cos {60}^{\circ }\left(\cos {24}^{\circ }\cos {84}^{\circ }\cos {36}^{\circ }\right)=\frac{1}{2}\left(\cos {12}^{\circ }\cos \right({60}^{\circ }+{12}^{\circ }\left)\cos \right({60}^{\circ }-{12}^{\circ }\left)\right)\left(\cos {24}^{\circ }\cos \right({60}^{\circ }+{24}^{\circ }\left)\cos \right({60}^{\circ }-{24}^{\circ }\left)\right)=\frac{1}{2}\left[\frac{1}{4}\cos \right(3\times {12}^{\circ })\times \frac{1}{4}\cos (3\times {24}^{\circ }\left)\right]=\frac{1}{32}\left(\cos {36}^{\circ }\cos {72}^{\circ }\right)=\frac{1}{32}\times \frac{\sqrt{5}+1}{4}\times \frac{\sqrt{5}-1}{4}=\frac{1}{128}$