$\begin{align}  & {{(\sin B-\cos B)}^{2}}+{{(\sin C-\cos C)}^{2}}-{{(\sin A-\cos A)}^{2}} \\ & =\left[ {{\sin }^{2}}B+{{\cos }^{2}}B-2\sin B\cos B+{{\sin }^{2}}C+{{\cos }^{2}}C-2\sin C\cos C-({{\sin }^{2}}A+{{\cos }^{2}}A-2\sin A\cos B) \right] \\ & =\left[ 1-\sin 2B+1-\sin 2C-1+\sin 2A \right] \\ & =\left[ 1-(\sin 2B+\sin 2C)-\sin 2A \right] \\ & =1-2\sin (B+C)\cos (B-C)+\sin 2A \\ & \text{Note: }{{180}^{\circ }}-A=B+C \\ & =1-2\sin ({{180}^{\circ }}-A)\cos (B-C)+\sin 2A \\ & =1-2\sin A\cos (B-C)+2\sin A\cos A \\ & =1-[2\sin A\cos (B-C)-2\sin A\cos A] \\ & =1-2\sin A[\cos (B-C)-\cos A] \\ & =1-2\sin A[\cos (B-C)-\cos ({{180}^{\circ }}-(B+C)] \\ & =1-2\sin A[\cos (B-C)+\cos (B+C)] \\ & =1-2\sin A[\left[ 2\cos \tfrac{B+C+B-C}{2}\cos \tfrac{B+C-B+C}{2} \right] \\ & =1-4\sin A\cos B\cos C \\\end{align}$