The number of solution of the equation $sin 5 x cos 3 x = sin 6 x cos 2 x$ in the interval $[0, π]$ is

2

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3

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4

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5

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Solution

The correct option is A $2$
$sin 5 x cos 3 x = sin 6 x cos 2 x$
$sin θ cos ϕ = \frac{1}{2} [sin (θ + ϕ) + sin (θ - ϕ)]$
$\frac{1}{2} [sin 8 x + sin 2 x] = \frac{1}{2} [sin 8 x + sin 4 x]$
$\frac{1}{2} sin 2 x = \frac{1}{2} sin 4 x$
$sin 4 x - sin 2 x = 0$
$2 sin 2 x cos 2 x - sin 2 x = 0$
$sin 2 x [2 cos 2 x - 1] = 0$
$sin 2 x = 0$ $2 cos 2 x - 1 = 0$
$2 x = 0$ $cos 2 x = \frac{1}{2}$
$x = 0$ $2 x = \frac{π}{3}$
$x = \frac{π}{6}$
$t w o$ solutions.

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