First slide

Trigonometric transformations

Question

In a $Δ A B C, \sin^{4} A + \sin^{4} B + \sin^{4} C =$

Moderate

A

$\frac{3}{2} + 2 \cos A \cos B \cos C + \frac{1}{2} \cos 2 A \cos 2 B \cos 2 C$
B

$2 \cos A \cos B \cos C + \frac{1}{2} \cos 2 A \cos 2 B \cos 2 C$
C

$2 \cos A \cos B \cos C + \frac{1}{2} \cos 2 A \cos 2 B$
D

$2 \cos A \cos B \cos C + \cos 2 A \cos 2 B$

Solution

L.H.S $= {(\frac{1 - \cos 2 A}{2})}^{2} + {(\frac{1 - \cos 2 B}{2})}^{2} + {(\frac{1 - \cos 2 C}{2})}^{2}$
$= \frac{3}{4} + \frac{1}{4} (\cos^{2} 2 A + \cos^{2} 2 B + \cos^{2} 2 C)$
$- \frac{1}{2} (\cos 2 A + \cos 2 B + \cos 2 C)$

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