Sin²+2sinxcosx=3cos²x

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To solve this equation, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1.

First, let's rewrite the equation using this identity:
sin^2(x) + 2sin(x)cos(x) = 3cos^2(x)
sin^2(x) + 2sin(x)cos(x) = 3(1 - sin^2(x))

Now, expand the right side of the equation:
sin^2(x) + 2sin(x)cos(x) = 3 - 3sin^2(x)

Rearrange the terms and simplify:
sin^2(x) + 2sin(x)cos(x) + 3sin^2(x) = 3
4sin^2(x) + 2sin(x)cos(x) = 3

Now, we can rewrite the equation in terms of only one trigonometric function (sin or cos). Let's rewrite the cosine term using the identity sin^2(x) + cos^2(x) = 1:
4(1 - cos^2(x)) + 2sin(x)cos(x) = 3
4 - 4cos^2(x) + 2sin(x)cos(x) = 3

Rearrange the terms:
4 - 4cos^2(x) + 2sin(x)cos(x) - 3 = 0
-4cos^2(x) + 2sin(x)cos(x) + 1 = 0

Now, factor out a -1 from the equation:
-1(4cos^2(x) - 2sin(x)cos(x) - 1) = 0

Now, you can solve for the value of x.