Question

If $x=\cos {10}^o\cos {20}^o\cos {40}^o$, then the value of $x$ is

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

Answer 1

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

$x=\cos 10°\cos 20°\cos 40°$

$\cos 10°\cos 20°\cos 40°=\cos 10°\cos 2\times 10°\cos 2^2\times 10°$

$A=10°$

$\cos A\cos 2A\cos 2^{2}A\dots \cos 2^{n-1}A=\frac{\sin 2^{n}A}{2^n\sin A}$

$\frac{\sin 80°}{8\sin 10°}$

$=\frac{\sin (90°-10°)}{8\sin 10°}$

$=\frac{1}{8}\times \frac{\cos 10°}{\sin 10°}$

$=\frac{1}{8}\cot 10°$