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Question

The number of common solution(s) of the trigonometric equations cos2x+(13)=(23)cosx and sin3x=2sinx, satisfying the inequality 3tanx10 in [0,5π] is

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Solution

2cos2x1+13=(23)cosx
cosx(cosx1)+3(cosx1)=0
(cosx1)(2cosx+3)=0
cosx=1 or cosx=32

sin3x=2sinx
3sinx4sin3x=2sinx
sinx=0 or 34sin2x=2

4sin2x=1
sinx=±12

Common solutions in [0,5π] are 0,2π,4π,5π6,7π6,17π6,19π6,29π6

If tanx13
x[π6+nπ,π2+nπ];nZ

So, common solutions x=7π6,19π6

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