Question

The value of $\frac{\cos {15}^{\circ }\cos {30}^{\circ }\cos {45}^{\circ }\cos {60}^{\circ }\cos {75}^{\circ }}{\sin {10}^{\circ }\sin {30}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }}$ is

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

Answer 1

The correct option is A $\sqrt{\frac{3}{2}}$

$\frac{\cos {15}^{\circ }\cos {30}^{\circ }\cos {45}^{\circ }\cos {60}^{\circ }\cos {75}^{\circ }}{\sin {10}^{\circ }\sin {30}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }}=\frac{\cos {30}^{\circ }\cos {60}^{\circ }\cos {15}^{\circ }\cos {45}^{\circ }\cos {75}^{\circ }}{\sin {30}^{\circ }\sin {10}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }}=\frac{\cos {30}^{\circ }\cos {60}^{\circ }\frac{1}{4}\cos (3\times {15}^{\circ })}{\sin {30}^{\circ }\frac{1}{4}\sin (3\times {10}^{\circ })}=\frac{\cos {30}^{\circ }\cos {60}^{\circ }\cos \left({45}^{\circ }\right)}{{\sin }^2{30}^{\circ }}$
$=\sqrt{\frac{3}{2}}$