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.
