Why does ##cos(90 - x) = sin(x)## and ##sin(90 - x) = cos(x)##?

Note that the image below is only for ##x## in Q1. If you wish you should be able to draw it with ##x## in any quadrant.

Definition of ##sin(x)## (side opposite angle x)/hypotenuse

Definition of ##cos(90^o -x)## (side adjacent to angle (##90^o-x##))/hypotenuse

but (side opposite angle x) ##=## (side adjacent to angle (##90^o-x##)

Therefore ##sin(x) = cos(90^o -x)##

Similarly ##cos(x) = sin(90^o - x)##