5sin^2x+3sinxcosx-2cos^2x=3

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To simplify this expression, we can use the trigonometric identity:

sin^2x + cos^2x = 1

Starting with the given expression:

5sin^2x + 3sinxcosx - 2cos^2x = 3

Rearranging the terms:

5sin^2x + 3sinxcosx - 2cos^2x - 3 = 0

Now we will substitute sin^2x with (1 - cos^2x):

5(1 - cos^2x) + 3sinxcosx - 2cos^2x - 3 = 0

Expand the terms:

5 - 5cos^2x + 3sinxcosx - 2cos^2x - 3 = 0

-7cos^2x + 3sinxcosx + 2 = 0

Now we will use the trigonometric identity sin2x = 2sinxcosx to rewrite the expression:

-7(1 - sin^2x) + 3sinx(2sinx) + 2 = 0

-7 + 7sin^2x + 6sin^2x + 2 = 0

13sin^2x - 7 = 0

Now divide by 13:

sin^2x = 7/13

This is the simplified form of the given expression.