Explanation:
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
x = theta
y = 90°
cos(90) = 0
sin(90) = 1
cos(x)×0 - sin(x)×1 = - sin(x)
remember slopes of lines ?
and how the coordinates and the slope changed for a perpendicular line ?
the slope is y/x, and for the perpendicular line this turned into -x/y.
the same happens here.
sin(x) and cos(x) are the x and y coordinates of a point on the circle with the angle x to the horizontal axis.
if you add 90° to the angle, you create a perpendicular line or angle, and the coordinates follow the slope rules : cos turns into sin and vice versa, and the sign has to flip.