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Question
If A + B + C =π. Prove that
cos 2A + cos 2B + cos 2C = -1 -4 cos A cos B cos C.
If A + B + C =π. Prove that
cos 2A + cos 2B + cos 2C = -1 -4 cos A cos B cos C.
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Solution
cos2A+cos2B+cos2C
=2cos(A+B)cos(A−B)+2cos2C−1
=2cos(π−C)cos(A−B)+2cos2C−1
=−2cosC cos(A−B)+2cos2C−1
=−2cosC [cos(A−B)−cosC]−1
= −1−2cos C[cos(A−B)−cos(π−(A+B))]
= −1−2cos C[cos(A−B)+cos(A+B)]
=−1−2cos C[2cos A cos B]
=−1−4cos A cos B cos C=RHS.
∴ cos 2A+cos 2B+cos 2C=−1−4cos A cos B cos C.
cos2A+cos2B+cos2C
=2cos(A+B)cos(A−B)+2cos2C−1
=2cos(π−C)cos(A−B)+2cos2C−1
=−2cosC cos(A−B)+2cos2C−1
=−2cosC [cos(A−B)−cosC]−1
= −1−2cos C[cos(A−B)−cos(π−(A+B))]
= −1−2cos C[cos(A−B)+cos(A+B)]
=−1−2cos C[2cos A cos B]
=−1−4cos A cos B cos C=RHS.
∴ cos 2A+cos 2B+cos 2C=−1−4cos A cos B cos C.
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