4 sin2 х - cos х - 1 = 0

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To solve this equation, we can first use a trigonometric identity to replace the sin^2 x term.

The trigonometric identity sin^2 x + cos^2 x = 1 can be rearranged to give sin^2 x = 1 - cos^2 x.

Substitute sin^2 x = 1 - cos^2 x into the original equation:

4(1 - cos^2 x) - cos x - 1 = 0
Expanding:

4 - 4cos^2 x - cos x - 1 = 0
Rearranging:

4cos^2 x + cos x - 3 = 0
Now, let u = cos x:

4u^2 + u - 3 = 0
This is a quadratic equation that we can solve using the quadratic formula:

u = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 4, b = 1, and c = -3. Plug in these values:

u = (-(1) ± √((1)^2 - 4(4)(-3))) / 2(4)
u = (-1 ± √(1 + 48)) / 8
u = (-1 ± √49) / 8
u = (-1 ± 7) / 8

There are two possible solutions for u:

u = (7-1) / 8 = 6 / 8 = 3 / 4
u = (-7-1) / 8 = -8 / 8 = -1

Now, recall that u = cos x.

So, cos x = 3/4 or cos x = -1.

These are the solutions to the equation 4sin^2 x - cos x - 1 = 0.