Sin(A-B)/cosA.cosB + sin (B-C)/cosB.cosC +sin (C-A)/cosC.cosA …

${{\sin \left( {{\rm{A}} - {\rm{B}}} \right)} \over {\cos {\rm{A}}\cos {\rm{B}}}} + {{\sin \left( {{\rm{B}} - {\rm{C}}} \right)} \over {\cos {\rm{B}}\cos {\rm{C}}}} + {{\sin \left( {{\rm{C}} - {\rm{A}}} \right)} \over {\cos {\rm{C}}\cos {\rm{A}}}}$

= ${{\sin {\rm{A}}\cos {\rm{B}} - \cos {\rm{A}}\sin {\rm{B}}} \over {\cos {\rm{A}}\cos {\rm{B}}}} + {{\sin {\rm{B}}\cos {\rm{C}} - \cos {\rm{B}}\sin {\rm{C}}} \over {\cos {\rm{B}}\cos {\rm{C}}}} + {{\sin {\rm{C}}\cos {\rm{A}} - \cos {\rm{C}}\sin {\rm{A}}} \over {\cos {\rm{C}}\cos {\rm{A}}}}$

= $\eqalign{ & {{\sin {\rm{A}}\cos {\rm{B}}\cos {\rm{C}} - \cos {\rm{A}}\sin {\rm{B}}\cos {\rm{C}} + \cos {\rm{A}}\sin {\rm{B}}\cos {\rm{C}} - \cos {\rm{A}}\cos {\rm{B}}\sin {\rm{C}} + \sin {\rm{C}}\cos {\rm{A}}\cos {\rm{B}} - \cos {\rm{B}}\cos {\rm{C}}\sin {\rm{A}}} \over {\cos {\rm{A}}\cos {\rm{B}}\cos {\rm{C}}}} \cr & \cr}$

= ${0 \over {\cos {\rm{A}}\cos {\rm{B}}\cos {\rm{C}}}}$

= 0

Hence proved