Because $A:B:C=4:2:1$ and $A+B+C=\pi$, therefore $A=\frac{4}{7}\pi$, $B=\frac{2}{7}\pi$, and $C=\frac{1}{7}\pi$. By law of sines, the to-be-claimed relation is equivalent to $$\frac{1}{\sin{\frac{4}{7}\pi}}+\frac{1}{\sin{\frac{2}{7}\pi}}=\frac{1}{\sin{\frac{1}{7}\pi}}$$ or equivalently by multiplying $(\sin{\frac{4}{7}\pi}\sin{\frac{2}{7}\pi}\sin{\frac{1}{7}\pi})$ on both sides: $$\sin{\frac{1}{7}\pi}\sin{\frac{2}{7}\pi}+\sin{\frac{1}{7}\pi}\sin{\frac{4}{7}\pi}=\sin{\frac{2}{7}\pi}\sin{\frac{4}{7}\pi}$$ The left side equals \begin{align} &\sin{\frac{1}{7}\pi}\Big(\sin{\frac{2}{7}\pi}+\sin{\frac{4}{7}\pi}\Big)\\ &=\sin{\frac{1}{7}\pi}(2\sin{\frac{3}{7}\pi}\cos{\frac{1}{7}\pi})\\ &=(2\sin{\frac{1}{7}\pi}\cos{\frac{1}{7}\pi})\sin{\frac{3}{7}\pi}\\ &=\sin{\frac{2}{7}\pi}\sin{\frac{3}{7}\pi}\\ &=\sin{\frac{2}{7}\pi}\sin{\frac{4}{7}\pi} \end{align} Hence, the left side indeed equals the right side.