Cos2x-1=корень из 2 sin(5П/2-х)

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To solve this equation, we will use trigonometric identities to simplify it. Let's start by simplifying the right side using the angle difference identity:

sin(5π/2 - x) = sin(5π/2)cos(x) - cos(5π/2)sin(x)
sin(5π/2) = 1 and cos(5π/2) = 0
sin(5π/2 - x) = 1cos(x) - 0sin(x) = cos(x)

Now, the equation becomes:

cos(2x) - 1 = √2sin(5π/2 - x)
cos(2x) - 1 = √2cos(x)

Now, we can use the double angle identity for cosine to further simplify this equation:

2cos^2(x) - 1 - 1 = √2cos(x)
2cos^2(x) - 2 = √2cos(x)

Now, we have a quadratic equation in terms of cos(x):

2cos^2(x) - √2*cos(x) - 2 = 0

Solving this quadratic equation will give us the values of cos(x), and then we can find the corresponding values of x.