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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
$\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}=$
  • A
    $\frac{1}{16}$
  • B
    $0$
  • C
    $\frac{-1}{8}$
  • ✓
    $\frac{-1}{16}$
✓

Answer

Correct option: D.
$\frac{-1}{16}$
(D)
$\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$
$\cos \alpha \cdot \cos 2 \alpha \cdot \cos 2^2 \alpha \cdot \cos 2^3 \alpha \ldots . . \cos 2^{n-1} \alpha$
$=\frac{\sin 2^n \alpha}{2^n \sin \alpha}, \text { if } \alpha \neq n \pi$
$=1, \text { if } \alpha=2 n \pi$
$=-1, \text { if } \alpha=(2 n+1) \pi$
$ =\frac{\sin \frac{2^4 \pi}{5}}{2^4 \sin \frac{\pi}{5}}=\frac{\sin \frac{16 \pi}{5}}{16 \sin \frac{\pi}{5}} $
$=\frac{\sin \left(3 \pi+\frac{\pi}{5}\right)}{16 \sin \frac{\pi}{5}}$
$=\frac{-\sin \frac{\pi}{5}}{16 \sin \frac{\pi}{5}}=-\frac{1}{16}$

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