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Question
If A + B + C = π, then cos2 A + cos2 B + cos2 C is equal to ______.
Options
1 – cos A cos B cos C
1 – 2 sin A sin B sin C
1 – sin A sin B sin C
1 – 2 cos A cos B cos C
MCQ
Fill in the Blanks
Solution
If A + B + C = π, then cos2 A + cos2 B + cos2 C is equal to 1 – 2 cos A cos B cos C.
Explanation:
We have cos2 A + cos2 B + cos2 C
= cos2 A + (1 – sin2 B) + cos2 C
= (cos2 A – sin2 B) + cos2 C + 1
= cos(A + B) cos(A – B) + cos2 C + 1
= cos(π – C) cos(A – B) + cos2 C + 1
= – cos C cos(A – B) + cos2 C + 1
= – cos C[cos(A – B) – cos C] + 1
= – cos C[cos(A – B) – cos{π – (A + B)}] + 1
= – cos C[cos(A – B) + cos(A + B)] + 1
= – cos C(2 cos A cos B) + 1
= 1 – 2 cos A cos B cos C
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Factorization Formulae - Trigonometric Functions of Angles of a Triangle
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