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

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To solve the equation 2sin^4(2x) + cos(4x) - 1 = 0, we can use trigonometric identities to simplify the expression.

Recall the following trigonometric identities:

Let's start by substituting sin^2(2x) = 1 - cos^2(2x) into the equation:

2(1 - cos^2(2x))^2 + cos(4x) - 1 = 0
2(1 - 2cos^2(2x) + cos^4(2x)) + cos(4x) - 1 = 0
2 - 4cos^2(2x) + 2cos^4(2x) + cos(4x) - 1 = 0
2cos^4(2x) - 4cos^2(2x) + cos(4x) + 1 = 0

Now, let's substitute cos(2x) = 2cos^2(x) - 1 into the equation:

2(2cos^2(x) - 1)^2 - 4(2cos^2(x) - 1) + cos(4x) + 1 = 0
2(4cos^4(x) - 4cos^2(x) + 1) - 8cos^2(x) + 4 + cos(4x) + 1 = 0
8cos^4(x) - 8cos^2(x) + 2 - 8cos^2(x) + 5 + cos(4x) = 0
8cos^4(x) - 16cos^2(x) + cos(4x) + 7 = 0

Now, we have the simplified equation 8cos^4(x) - 16cos^2(x) + cos(4x) + 7 = 0. This equation is in terms of cos(x), and we can use trigonometric identities to further simplify or solve it.