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AM,GM,HM Inequality

In Δ ABC ...

Question

In $Δ$ $A B C$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively.
The value of $a cos A + b cos B + c cos C$ is equal to?

2 a sin B sin C

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2 b sin C sin A

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2 c sin A sin B

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None of the above

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Solution

The correct options are
A $2 a sin B sin C$
B $2 b sin C sin A$
C $2 c sin A sin B$
We know that according to Sine law;
$\frac{a}{sin A} = \frac{a}{sin B} = \frac{c}{sin C} = 2 R$
We have;
$a cos A + b cos B + c cos C$
$= 2 R sin A . cos A + 2 R sin B . cos B + 2 R sin C . cos C$
$= R (sin 2 A + sin 2 B + sin 2 C)$
$= R \times 4 sin A . sin B . sin C$
$= 2. (2 R sin A) sin B . sin C = 2 a sin B . sin C$
$= 2. (2 R sin B) sin A . sin C = 2 b sin A . sin C$
$= 2. (2 R sin C) sin A . sin B = 2 c sin A . sin C$

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