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All we need to do is to substitute x by pi/4 in the expresison of the function:
f(pi/4) = [cos (pi/4)]^2 - sin 2*(pi/4)
f(pi/4) = [cos (pi/4)]^2 - sin (pi/2)
We know that cos pi/4 = (sqrt2)/2 and sin (pi/2) = 1.
f(pi/4) = [(sqrt2)/2]^2 - 1
f(pi/4) = 1/2 - 1
f(pi/4) = -1/2
The requested value of the function f(x) = [cos (x)]^2 - sin (2x), if x = pi/4, is: f(pi/4) = -1/2.
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