Question

If A + B = C, then cos² A + cos² B + cos² C – 2 cos A cos B cos C is equal to ______.

पर्याय

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Answer 1

If A + B = C, then cos² A + cos² B + cos² C – 2 cos A cos B cos C is equal to 1.

Explanation:

cos² A + cos² B + cos² C

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

= ${\displaystyle 1+\frac{1}{2}(\cos 2A+\cos 2B)+{\cos }^{2}C}$

= ${\displaystyle 1+\frac{2}{2}[\cos (A+B)\cos (A-B)+{\cos }^{2}C}$

= 1 + cos C cos (A – B) + cos C cos (A + B)

= 1 + cos C [cos (A – B) + cos (A + B)]

= 1 + 2 cos C cos B cos A

${\displaystyle \Rightarrow }$ cos² A + cos² B + cos² C – 2 cos A cos B cos C = 1