Q. $\tan{6{^\circ}}\tan{42{^\circ}}\tan{66{^\circ}}\tan{78{^\circ}}\$is equal to
A) 1
B) $\frac{1}{2}$
C) $\frac{1}{4}$
D) $\frac{1}{8}$
Explanation
(a)
$\tan{6{^\circ}\tan{42{^\circ}\tan{66{^\circ}\tan{78{^\circ}}}}}$
$= \tan{6{^\circ}\tan{(60{^\circ} - 18{^\circ})\tan{(60{^\circ} + 6{^\circ})\tan(60{^\circ} + 18{^\circ})}}}$
$= \frac{\begin{array}{r}
\tan{6{^\circ}\tan{(60{^\circ} + 6{^\circ})\tan{18{^\circ}}}} \\
\tan{(60{^\circ} - 18{^\circ})\tan{(60{^\circ} + 18{^\circ})}}
\end{array}}{\tan{18{^\circ}}}$
$= \frac{\tan{6{^\circ}\tan{(60{^\circ} + 6{^\circ})\tan{(3 \times 18{^\circ})}}}}{\tan{18{^\circ}}}$
$= \frac{\tan{6{^\circ}\tan{(60{^\circ} - 6{^\circ})\tan{(60{^\circ} + 6{^\circ})}}}}{\tan{18{^\circ}}}$
$= \frac{\tan{18{^\circ}}}{\tan{18{^\circ}}} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.