(1)

\begin{align*} a\cos A+b\cos B+c\cos C & =2R\left(\sin A\cos A+\sin B\cos B+\sin C\cos C\right)\\ & =R\left(\sin\left(2A\right)+\sin\left(2B\right)+\sin\left(2C\right)\right)\\ & =4R\sin A\sin B\sin C\\ & =4R\frac{bc\sin Aca\sin Bab\sin C}{a^{2}b^{2}c^{2}}\\ & =4R\frac{8S^{3}}{a^{2}b^{2}c^{2}}\\ & =\frac{8S^{2}}{abc}\cmt{\because S=\frac{abc}{4R}} \end{align*}

(2)

\begin{align*} a\sin A+b\sin B+c\sin C & =a\cdot\frac{a}{2R}+b\cdot\frac{b}{2R}+c\cdot\frac{c}{2R}\\ & =\frac{a^{2}+b^{2}+c^{2}}{2R}\\ & =\frac{2S\left(a^{2}+b^{2}+c^{2}\right)}{abc}\cmt{\because S=\frac{abc}{4R}} \end{align*}