Question

If $x=\cos {10}^{\circ }\cos {20}^{\circ }\cos {40}^{\circ }$ then $x$ equals to

• A. $\frac{1}{4}\tan {10}^{\circ }$
• B. $\frac{1}{8}\cot {10}^{\circ }$
• C. $\frac{1}{8}\csc {10}^{\circ }$
• D. $\frac{1}{8}\sec {10}^{\circ }$

Answer 1

The correct option is B $\frac{1}{8}\cot {10}^{\circ }$

$x=\cos {10}^o\cos {20}^o\cos {40}^o$

$=\frac{1}{2}\times 2\cos {10}^o\cos {20}^o\cos {40}^o$

$=\frac{1}{2\sin {10}^o}\times 2\sin {10}^o\cos {10}^o\cos {20}^o\cos {40}^o$

$=\frac{1}{2\sin {10}^o}\times \sin {20}^o\cos 20\cos {40}^o$

$=\frac{1}{4\sin {10}^o}\times 2\sin {20}^o\cos 20\cos {40}^o$

$=\frac{1}{4\sin {10}^o}\times \sin {40}^o\cos {40}^o$

$=\frac{1}{8\sin {10}^o}\times 2\sin {40}^o\cos {40}^o$

$=\frac{1}{8\sin {10}^o}\times 2\sin {40}^o\cos {40}^o$

$=\frac{1}{8\sin {10}^o}\times \sin {80}^o$

$=\frac{\sin ({90}^o-{10}^o)}{8sin{10}^o}$

$=\frac{\cos {10}^o}{8\sin {10}^o}$

$=\frac{1}{8}\cot {10}^o$