Sin^2 (2x)+sin^2(x)=1.5

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To solve this equation, we can first use the double angle identity for sin(2x):

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

Substitute this into the original equation:

(1 - cos(4x))/2 + sin^2(x) = 1.5

Multiply through by 2 to eliminate the fraction:

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

Now we can use the Pythagorean trigonometric identity sin^2(x) + cos^2(x) = 1:

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

Simplify:

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

Combine like terms:

3sin^2(x) = 3

Divide by 3:

sin^2(x) = 1

Taking the square root of both sides:

sin(x) = ±1

Since sin(x) cannot equal ±1 simultaneously, there must be no solution to this equation.