L=1/sinA-sinA; M=1/cosA-cosA; L^2=1/sin^2(A)+sin^2(A)-2; M^2=1/cos^2(A)+cos^2(A)-2.
L^2+M^2=(1/sin^2(A)cos^2(A))-3, so L^2+M^2+3=1/sin^2(A)cos^2(A). (sin^2(A)+cos^2(A)=1)
L=cos^2(A)/sinA; M=sin^2(A)/cosA; LM=sinAcosA; L^2M^2=sin^2(A)cos^2(A).
Therefore, L^2M^2(L^2+M^2+3)=sin^2(A)cos^2(A)/sin^2(A)cos^2(A)=1 QED