(1+2sinx)sinx=sin2x+cosx

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вопрос

To solve the equation (1 + 2sinx)sinx = sin2x + cosx, we need to first expand the left side of the equation.

Expanding the left side:
(1 + 2sinx)sinx = sinx + 2sin^2(x) = sinx + 2sinxsinx = sinx + 2sin^2(x)

Now, let's expand the right side of the equation.
sin2x + cosx = 2sinxcosx + cosx

Now, equating the left and right sides of the equation:
sinx + 2sin^2(x) = 2sinxcosx + cosx

To simplify further, we can use trigonometric identities:
sin^2(x) = 1 - cos^2(x)
2sinxcosx = sin2x

Substitute the identities into the equation:
sinx + 2(1 - cos^2(x)) = sin2x + cosx
sinx + 2 - 2cos^2(x) = sin2x + cosx

Rearrange the terms:
-2cos^2(x) + sinx - cosx = sin2x - sinx + 2

Now, we can try to solve for x by simplifying and rearranging the equation further.