We know that the angles of a right triangle must sum to #pi# radians. Since the right angle is #pi/2# radians, the other two angles must sum to #pi/2# radians. Therefore #pi/2# minus the adjacent angle must be the opposite angle. Another way of saying this is #cos(pi/2-x)=sinx#, so
#cos(pi/2-x)*csc(-x)=sinx*csc(-x)#.
The definition of #cscx# says that #cscx=1/sinx#, so
#sinx*csc(-x)=sinx/sin(-x)#
We know that #sinx# is an odd function so #sin(-x)=-sinx# and we can write
#sinx/sin(-x)=-sinx/sinx=-1#.
However, this breaks down when #x# is a multiple of #pi# because #csc(npi)# is not defined.
