4sin2x - 3sinxcosx + 5cos2x = 3

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To simplify the given trigonometric equation, we can use the double angle formulas to express sin2x and cos2x in terms of sinx and cosx.

sin2x = 2sinxcosx
cos2x = cos^2x - sin^2x
cos2x = cos^2x - (1 - cos^2x) [using sin^2x + cos^2x = 1]
cos2x = 2cos^2x - 1

Now we can substitute sin2x = 2sinxcosx and cos2x = 2cos^2x - 1 back into the original equation:

4(2sinxcosx) - 3sinx cosx + 5(2cos^2x - 1) = 3
8sinxcosx - 3sinx cosx + 10cos^2x - 5 = 3
5sinxcosx + 10cos^2x - 8 = 3
5sinxcosx + 10cos^2x = 11

Now we can use the double angle formula for sin2x to express 5sinxcosx in terms of sin2x:

sin2x = 2sinxcosx
5sinxcosx = 5/2sin2x

Substitute 5sinxcosx with 5/2sin2x:

5/2sin2x + 10cos^2x = 11

Now we can use cos2x = 2cos^2x - 1 to further simplify:

5/2sin2x + 10(2cos^2x - 1) = 11
5/2sin2x + 20cos^2x - 10 = 11
5/2sin2x + 20cos^2x = 21

Since we have already simplified the equation as much as possible, the final simplified equation is:

5/2sin2x + 20cos^2x = 21