Sinx x cosx+2sin^2x=cos^2x

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To solve this trigonometric equation, we need to use trigonometric identities to simplify the expression on both sides of the equation.

Starting with the left-hand side:

sin(x) cos(x) + 2 sin^2(x)
= sin(x) cos(x) + 2 (1 - cos^2(x)) [using the identity sin^2(x) = 1 - cos^2(x)]
= sin(x) cos(x) + 2 - 2cos^2(x)
= sin(x) cos(x) + 2 - 2cos^2(x)

Now, we'll simplify the right-hand side of the equation:

cos^2(x)
= 1 - sin^2(x) [using the identity cos^2(x) = 1 - sin^2(x)]
= 1 - (1 - cos^2(x))
= 1 - 1 + cos^2(x)
= cos^2(x)

Now, the equation becomes:

sin(x) * cos(x) + 2 - 2cos^2(x) = cos^2(x)

Rearranging the terms gives:

sin(x) * cos(x) + 2 = 3cos^2(x)

Now, substituting the cosine identity sin(x) = √(1 - cos^2(x)) to the left-hand side:

√(1 - cos^2(x)) * cos(x) + 2 = 3cos^2(x)

Expanding the left side gives:

cos(x)√(1 - cos^2(x)) + 2 = 3cos^2(x)

At this point, you can square both sides, solve for 0, and adjust the equations to match your preferred form.