First slide

Trigonometric transformations

Question

If $A + B + C = 3 π / 2, then \cos 2 A + \cos 2 B + \cos 2 C$ is equal to

Moderate

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
D

1 - 4 sin A sin B sin C

Solution

$\begin{array}{l} \cos 2 A + \cos 2 B + \cos 2 C \\ = 2 \cos (A + B) \cos (A - B) + \cos 2 C \\ = 2 \cos (\frac{3 π}{2} - C) \cos (A - B) + \cos 2 C \\ = - 2 \sin Ccos (A - B) + 1 - 2 \sin^{2} C \\ = 1 - 2 \sin C (\cos (A - B) + \sin C) \\ = 1 - 2 \sin C {\cos (A - B) + \sin [3 π / 2 - (A + B)]} \\ = 1 - 2 \sin C [\cos (A - B) - \cos (A + B)] \\ = 1 - 4 \sin Asin Bsin C \end{array}$

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