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If $A + B + C = 270 °$ then $\cos 2 A + \cos 2 B + \cos 2 C + 4 \sin A \sin B \sin C =$

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answer is B.

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Detailed Solution

A + B + C = 270 °

then

\cos 2 A + \cos 2 B + \cos 2 C + 4 \sin A \sin B \sin C

$= 2 \cos (A + B) . \cos (A - B) + 1 - 2 \sin^{2} C + 4 \sin A . \sin B . \sin C$

$= - 2 \sin C . \cos (A - B) - 2 \sin^{2} C + 1 + 4 \sin A . \sin B . \sin C$

$= - 2 \sin C . (\cos (A - B) + \sin C) + 1 + 4 \sin A . \sin B . \sin C$

$= - 2 \sin C . (\cos (A - B) - \cos (A + B)) + 1 + 4 \sin A . \sin B . \sin C$

$= - 2 \sin C (2 \sin A . \sin B) + 1 + 4 \sin A . \sin B . \sin C$

$= - 4 \sin A . \sin B . \sin C + 1 + 4 \sin A . \sin B . \sin C$

$= 1$

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