If A+B+C=3 2 , then cos 2A + cos 2B + cos 2C = - Vidyadip

MCQ

If $A+B+C=\frac{3 \pi}{2}$, then
cos 2A + cos 2B + cos 2C =

A
1 - 4 cos A cos B cos C
B
4 sin A sin B sin C
C
1 + 2 cos A cos B cos C
✓
1 - 4 sin A sin B sin C

✓

Answer

Correct option: D.

1 - 4 sin A sin B sin C

(D)
$\cos 2 A+\cos 2 B+\cos 2 C$
$=2 \cos (A+ B ) \cos ( A - B )+\cos 2 C$
$-2 \cos \left(\frac{3 \pi}{2}-C\right) \cos (A-B)+\cos 2 C$$\ldots\left[\because A+B=\frac{3 \pi}{2}-C\right]$
$=-2 \sin C \cos (A-B)+1-2 \sin ^2 C$
$-1-2 \sin C(\cos (A-B) \mid \sin C)$
$=1-2 \sin C\{\cos (A-B)-\cos (A+B)\}$
$=1-4 \sin A \sin B \sin C$
Trick : Check by assuming $A = B = C =\frac{\pi}{2}$

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