Draw a circle of radius 1 centre O, and draw two diameters at right angles, AOB and COD, dividing the unit circle into quadrants, COB, BOD, DOA, AOC. Draw a radius OX at angle x to OB. Also draw another radius OY at angle x to OC. Angles YOC and XOB=x. Also, YOB=90+x=(pi)/2+x.
Drop a perpendicular XP from X on to OB. OP=cosx.
Drop a perpendicular YQ to AO. SinYOA=sinYOB=sin((pi)/2+x)=YQ. Right-angled triangles YOQ, YOC and XOP are congruent and YQ=OP=cosx=sin((pi)/2+x).