Solve the equation algebraically for x where 0 </ x <2pi.

sin (x-(pi/2)) = cos^2 x

first of all

sin(x- pi/2) = -cosx
(you can get this by using the sin(A-B) expansion
so
sin (x-(pi/2)) = cos^2 x
-cosx = cos^2 x
cos^2 x + cosx = 9
cosx(cosx + 1) = 0
cosx = 0 or cosx = -1

so x = pi/2, pi, 3pi/2 (I looked at the graph of cosx)