Question

If $x=\cos 10°\cos 20°\cos 40°$, then the value of $x$ is:

• A. $\frac{1}{4}\tan 10°$
• B. $\frac{1}{8}\cot 10°$
• C. $\frac{1}{8}\mathrm{cosec}10°$
• D. $\frac{1}{8}\sec 10°$

Answer 1

The correct option is B

$\frac{1}{8}\cot 10°$

Explanation for the correct option.

$\begin{array}{rcl}x& =& \cos 10°\cos 20°\cos 40°\\ & =& \cos 10°\cos \left(2\times 10°\right)\cos \left(2^2\times 10°\right)\\ & =& \frac{\sin \left(2^3\times 10°\right)}{2^3\sin 10°}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{A}\mathbf{·}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\left(2A\right)\mathbf{·}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{·}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\left(2^{n-1}A\right)\mathbf{=}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\left(2^{n}A\right)}{{\mathbf{2}}^{\mathbf{n}}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{A}}\mathbf{]}\\ & =& \frac{\sin \left(80°\right)}{8\sin 10°}\\ & =& \frac{\sin \left(90°-10°\right)}{8\sin 10°}\\ & =& \frac{\cos 10°}{8\sin 10°}\mathbf{}\mathbf{}\left[\because sin\left(90°-\theta \right)=cos\theta \right]\\ & =& \frac{1}{8}\cot 10°\end{array}$

Hence, option B is correct.