The number of solutions of equation, $ \sin 5 x \cos 3 x=\sin 6 x \cos 2 x $, in the interval $ [0, \pi] $ are

Question

The number of solutions of equation, $ \sin 5 x \cos 3 x=\sin 6 x \cos 2 x $, in the interval $ [0, \pi] $ are, $ 3 $, $ 4 $, $ 5 $, $ 6 $

MARKS App · Accepted Answer

Given, sin 5 x cos 3 x = sin 6 x cos 2 x The given equation can be written as 2 sin 5 x cos 3 x = 2 sin 6 x cos 2 x ⇒ sin 8 x + sin 2 x = sin 8 x + sin 4 x ; ∵ 2 sin A cos B = sin A + B + sin A - B ⇒ sin 2 x - sin 4 x = 0 ⇒ - 2 sin x cos 3 x = 0 , Hence, sin x = 0 or cos 3 x = 0 . x = n π n ∈ I or 3 x = k π + π 2 k ∈ I Therefore, since x ∈ 0 , π , the given equation is satisfied, x = 0 , π , π 6 , π 2 or 5 π 6 . Hence number of solutions are 5 .

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