20^ 40^ 80^ =? - Vidyadip

MCQ

$\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}=?$

A
$\frac{1}{16}$
✓
$\frac{1}{8}$
C
$\frac{\sqrt{3}}{8}$
D
$\frac{\sqrt{3}}{16}$

✓

Answer

Correct option: B.

$\frac{1}{8}$

Given exp. $=\frac{1}{2}\left(2 \cos 20^{\circ} \cos 80^{\circ}\right) \cos 40^{\circ}$
$=\frac{1}{2}\left[\cos \left(80^{\circ}+20^{\circ}\right)+\cos \left(80^{\circ}-20^{\circ}\right)\right] \cos 40^{\circ}$
$=\frac{1}{2}\left[\left(\cos 100^{\circ}+\cos 60^{\circ}\right) \cos 40^{\circ}\right]$
$=\frac{1}{2}\left[\left(\cos 100^{\circ}+\frac{1}{2}\right) \cos 40^{\circ}\right]$
$=\frac{1}{4}\left(2 \cos 100^{\circ} \cos 40^{\circ}\right)+\frac{1}{4} \cos 40^{\circ}$
$\left.=\frac{1}{4} \cos \left(100^{\circ}+40^{\circ}\right)+\cos \left(100^{\circ}-40^{\circ}\right)\right]+\frac{1}{4} \cos 40^{\circ}$
$\left.=\frac{1}{4} \cos 140^{\circ}+\cos 60^{\circ}\right)+\frac{1}{4} \cos 40^{\circ}$
$=\frac{1}{4}\left(\cos 140^{\circ}+\cos 40^{\circ}\right)+\left(\frac{1}{4} \times \frac{1}{2}\right)$
$=\frac{1}{4}\left[\cos \left(180^{\circ}-40^{\circ}\right)+\cos 40^{\circ}\right]+\frac{1}{8}$
$=\frac{1}{4}\left(-\cos 40^{\circ}+\cos 40^{\circ}\right)+\frac{1}{8}$
$=\frac{1}{8}$

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