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Question

Number of common solution(s) of the trigonometric equations cos 2x+(13)=(23)cos x and sin 3x=2 sin x which satisfy the inequality 3tan x10 in [0,5π] is

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Solution

2cos2x1+13=(23)cos x

2cosx(cosx1)+3(cosx1)=0

(cosx1)(2cosx+3)=0

cos x=1 or cos x=32

sin 3x=2 sin x

3 sin x4 sin3 x=2 sin x

sin x=0 or 34sin2x=2

4sin2x=1

sin x=±12

common solution in [0,5π] is 0,2π,4π,5π6,7π6,17π6,19π6,29π6

If tan x13x[π6+nπ,π2+nπ];nI

So, common solution:x=7π6,19π6

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