You have the following equation:
cos(x)(cos(x) - 1) = 0
if the previous equation has solution, then, it is necessary either cos(x) = 0 or cos(x) - 1 = 0.
Then, you have two solutions for the given equations. Solve for the two previous equation, as follow:
cos(x) = 0 apply cos⁻¹ both sides
cos⁻¹(cos(x)) = cos⁻¹(0)
x = cos⁻¹(0)
x = 90° which is equivalent to pi/2 radians
cos(x) - 1 = 0 add 1 both sides
cos(x) = 1 apply cos⁻¹ both sides
cos⁻¹(cos(x)) = cos⁻¹(1)
x = cos⁻¹(1)
x = 0° which is equivalent to 0 radians
Hence, the solutions are x=0°, 90°, or x = 0 , pi/2.