MCQ
$sin \left(\frac{\pi}{14}\right) \ . \ sin \left(\frac{3\pi}{14}\right) \ . sin\left(\frac{5\pi}{14}\right) \ . \sin\left(\frac{7\pi}{14}\right) $ નું મુલ્ય ..........
- A$1$
- B$\frac{1}{4}$
- ✓$\frac{1}{8}$
- D$\frac{\sqrt{2}}{7}$
હવે, $ sin \left(\frac{\pi}{14}\right) . sin\left(\frac{3\pi}{14}\right). sin\left(\frac{5\pi}{14}\right) . sin \left(\frac{7\pi}{14}\right)$
$= cos \left(\frac{\pi}{2} - \frac{\pi}{14}\right) \cdot cos \left(\frac{\pi}{2} - \frac{3\pi}{14} \cdot \right)\cdot cos\left(\frac{\pi}{2}-\frac{5\pi}{14}\right)\cdot sin \left({\pi} - \frac{\pi}{2}\right)$
$= cos \frac{6\pi}{14} . cos \frac{4\pi}{14} . cos \frac{2\pi}{14} . 1$
$= cos \frac{3\pi}{7} . cos \frac{2\pi}{7} . cos \frac{\pi}{7}$
$= \frac{1}{2sin \frac{\pi}{7}} \left(2 sin \frac{\pi}{7} cos \frac{\pi}{7} cos \frac{2\pi}{7} cos \frac{3\pi}{7}\right)$
$= \frac{1}{2sin \frac{\pi}{7}} \left(sin \frac{2\pi}{7} cos \frac{2\pi}{7} cos \frac{3\pi}{7}\right)$
$= \frac{1}{4sin \frac{\pi}{7}} \left(2sin \frac{2\pi}{7} cos \frac{2\pi}{7} cos \frac{3\pi}{7}\right)$
$= \frac{1}{4sin \frac{\pi}{7}} \left(sin \frac{4\pi}{7} cos \frac{3\pi}{7}\right)$
$= \frac{1}{8sin \frac{\pi}{7}} \left(2sin \frac{4\pi}{7} sin \frac{3\pi}{7}\right)$
$=\frac{1}{8sin\frac{\pi}{7}}\left(sin\frac{7\pi}{7}+sin\frac{\pi}{7}\right)$
$= \frac{1}{8sin \frac{\pi}{7}} \left(0 + sin \frac{\pi}{7}\right)$
$ = \frac{1}{8}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.