Question 1197163
if you graph the equation, you can see what's going on.
the graph is shown below:
<img src = "http://theo.x10hosting.com/2022/101203.jpg">
you can see from the graph that when x = pi/2, y = arccos(cos(x)) = arccos(cos(pi/2)) = pi/2.
that would indicate that arccos(cos(x)) = x.
that's true when 0 < x < pi.
arccos(cos(-pi/2)) = pi/2 as well.
however, since the angle is -pi/2, then the coordinate pair is (-pi/2,pi/2) which indicates that arccos(x) = -x when -pi < x < 0
the equivalent positive angle of -x is equal to -pi/2 + 2pi = 3pi/2.
that would be 2pi + x because 2pi + (-pi/2) = 3pi/2.
this is very tough to see unless you can graph it.
the graphing software i used is at <a href = "https://www.desmos.com/calculator" target = "_blank">https://www.desmos.com/calculator</a>
it looks to me that your solution is selection C which is 2pi + x.


you didn't have -x as a solution.
you needed to now that -x was equivalent to 2pi - (-x) = 2pi + x.
arccos(cos(-pi/2) = pi/2.
arccos(cos(2pi+pi/2) = pi/2.


let me know if you have any questions.
theo