Given the formula sin 90+theta = cos theta and sin 180 - theta = sin theta
why sin 90+theta is not equal to sin 180-theta since both of them belong to 2nd quadrant in the unit circle.
You can do this with diagrams, but it's simpler (for me anyway) to do it according the sum of angle formulas. In general $\displaystyle sin( A + B ) = sin(A)~cos(B) + sin(B)~cos(A)$
$\displaystyle sin(90 + \theta) = sin(90)~cos( \theta ) + sin( \theta )~cos(90) = cos( \theta ) + 0 = cos(\theta)$
$\displaystyle sin(180 - \theta ) = sin(180)~cos( \theta ) - sin( \theta )~cos(180) = 0 + sin( \theta ) = sin( \theta )$ (Watch the double negative in the last step.)
-Dan