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**ticbol** 4cos^2(x) = 1

2cos(x) = +,-1

cos(x) = +,-1/2

When cos(x) = 1/2.

Cosine is positive in the 1st and 4th quadrants, so,

x = pi/3 and [2pi -pi/3 = 5pi/3] in the interval [0,2pi) or in one revolution.

In many revolutions, [1 revolution = 360 degrees = 2pi radians],

x = pi/3 +2k*pi, and 5pi/3 +2k*pi ---because the period or cycle of cosine is 2pi.

When cos(x) = -1/2.

Cosine is negative in the 2nd and 3rd quadrants, so,

x = [pi -pi/3 = 2pi/3] and [pi +pi/3 = 4pi/3] in the interval [0,2pi) or in one revolution.

In many revolutions,

x = 2pi/3 +2k*pi, and 4pi/3 +2k*pi

Hence,

x = pi/3 +2k*pi, 2pi/3 +2k*pi, 4pi/3 +2k*pi, 5pi/3 +2k*pi

Meaning, in all the 4 quadrants, x has a value or solution.

Meaning, the solution repeats every 180 degrees or every pi.

Meaning, the period or cycle here of cosine is 180deg or pi only. The effective period is no longer 360deg or 2pi.

Hence, x = pi/3 +k*pi and 2pi/3 +k*pi.