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Just a bit of trigonometry:
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
Let put a = x and b = -45°
cos(x - 45) = cos(x)cos(45°) - sin(x)sin(-45°)
cos(45°) = sin(45°) = -sin(45°) = sqrt(2)/2
Then it comes
cos(x-45°) = (sqrt(2)/2) x (cos(x) + sin(x))
that is
cos(x) + sin(x) = sqrt(2) x cos(x - 45°)
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