2 sin^2x = 2 sin2x -1

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To solve the equation 2sin^2(x) = 2sin(2x) - 1, we can first use the double angle identity for sine:

sin(2x) = 2sin(x)cos(x)

Substitute this back into the equation:

2sin^2(x) = 2(2sin(x)cos(x)) - 1
2sin^2(x) = 4sin(x)cos(x) - 1

Then, we can use the Pythagorean identity for sine and cosine:

sin^2(x) + cos^2(x) = 1
cos^2(x) = 1 - sin^2(x)

Substitute cos^2(x) = 1 - sin^2(x) back into the equation:

2sin^2(x) = 4sin(x)(1 - sin^2(x)) - 1
2sin^2(x) = 4sin(x) - 4sin^3(x) - 1

Rearranging terms:

4sin^3(x) - 2sin^2(x) + 4sin(x) - 1 = 0

Now, this cubic equation can be solved using numerical methods or factoring techniques.