What is x if cos(2x+pi/2)=cos(x-pi/2)?

To find x if
cos(2x+pi/2)=cos(x-pi/2).

If cosa = cos b, then a = 2npi+b
or 2npi-b.

Therefore  if cos(2x+pi/2)=cos(x-pi/2),
then

(i) 2x+pi/2 = 2npi+x-pi/2  or (ii) 2x+pi/2 =
2npi-x+p/2.

2x-x = 2npi-pi.

x
= (2n-1)pi, n = 0, 1, 2,.....

(ii) 2x+pi/2 = 2npi
-x+pi/2.

2x+x = 2npi.

3x=
2npi.

x = 2npi/3,  n =0, 1, 2,