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)##