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Sum of Trigonometric Ratios in Terms of Their Product

If sinx + s...

Question

If $sin x + sin y = 3 (cos y - cos x)$ , then the value of $sin 3 x + sin 3 y =$

A

- 1

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B

1

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C

2

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D

0

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Solution

The correct option is D $0$
$sin x + sin y = 3 (cos y - cos x)$
$\Rightarrow sin x + 3 cos x = 3 cos y - sin y . . . (1)$
Let $3 = a sin A$ and $1 = a cos A$
Therefore $(1)$ is
$a cos A sin x + a sin A cos x = a sin A cos y - a cos A sin y$
$\Rightarrow sin (A + x) = sin (A - y)$
$A + x = A - y \Rightarrow x = - y$
Now $sin 3 x + sin 3 y = sin 3 x + sin (- 3 x) = sin 3 x - sin 3 x = 0$

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