MCQ
If $A + B + C =\pi$, then $\cos ^2 A+\cos ^2 B-\cos ^2 C$ is equal to
- A$1-4 \sin A \cos B \sin C$
- B$1-2 \sin A \sin B \sin C$
- ✓$1-2 \sin A \sin B \cos C$
- D$1-4 \sin A \sin B \cos C$
✓
Answer
Correct option: C.
$1-2 \sin A \sin B \cos C$
(C)
$\cos ^2 A+\cos ^2 B-\cos ^2 C$
$=\frac{1}{2}(1+\cos 2 A)+\frac{1}{2}(1+\cos 2 B)$$-\frac{1}{2}(1+\cos 2 C )$
$=\frac{1}{2}+\frac{1}{2}(\cos 2 A+\cos 2 B-\cos 2 C )$
$=\frac{1}{2}+\frac{1}{2}(1-4 \sin A \sin B \cos C )$
$=1-2 \sin A \sin B \cos C$
$\cos ^2 A+\cos ^2 B-\cos ^2 C$
$=\frac{1}{2}(1+\cos 2 A)+\frac{1}{2}(1+\cos 2 B)$$-\frac{1}{2}(1+\cos 2 C )$
$=\frac{1}{2}+\frac{1}{2}(\cos 2 A+\cos 2 B-\cos 2 C )$
$=\frac{1}{2}+\frac{1}{2}(1-4 \sin A \sin B \cos C )$
$=1-2 \sin A \sin B \cos C$
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