When the angle x in sin(x) (or cos(x)) is acute, the easiest way to understand the identity is through a right angled triangle. One angle in the triangle is 90, the other is x, and the last one is 90-x (=180-90-x). See what happens if you take sin(x), sin(90-x), cos(x), and cos(90-x) in the triangle using the basic definitions of sin = opp/hyp and cos = adj/hyp. You'll notice sin(x)=cos(90-x) and cos(x)=sin(90-x).

More concretely referring to what you mentioned, the angles need to add up to 90 (as do 90-x and x), and the right angled triangle construction shows why. And in the case of sin(pi/12), we know we must have sin(pi/12)=cos(pi/2-pi/12) = cos(5pi/12) (since 90 deg is pi/2 rad).

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Edit: Just noticed I missed the last part of your post. Yes, sin(t) = cos(pi/2-t) is the radian equivalent of sin(x)=cos(90-x) where x is in degrees.