Cosx x/2-sin x/2=корень из 2/2

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To solve the equation cos(x/2) - sin(x/2) = √2/2, we can apply the half-angle identities for sine and cosine.

Recall that the half-angle identities are:

cos(x/2) = ±√((1 + cos(x))/2)
sin(x/2) = ±√((1 - cos(x))/2)

Substitute these identities into the equation:

±√((1 + cos(x))/2) - ±√((1 - cos(x))/2) = √2/2

We can simplify this equation by squaring both sides to eliminate the square roots:

(1 + cos(x))/2 - 2√((1 + cos(x))(1 - cos(x))/4) + (1 - cos(x))/2 = 2/4

Multiplying all terms by 4 to clear the denominators:

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

Simplify further:

2 + 2cos(x) - 4√(sin^2(x)/4) = 2

2 + 2cos(x) - 2|sin(x)| = 2

2cos(x) - 2|sin(x)| = 0

cos(x) - |sin(x)| = 0

From here, we can solve for x by considering different intervals where sin(x) is positive or negative. The solution may involve multiple solutions due to the absolute value function.