a: b*cosC+c*cosB
\(=AC\cdot\dfrac{CA^2+CB^2-AB^2}{2\cdot CA\cdot CB}+AB\cdot\dfrac{BA^2+BC^2-AC^2}{2\cdot BA\cdot BC}\)
\(=\dfrac{CA^2+CB^2-AB^2+BA^2+BC^2-AC^2}{2\cdot CB}=\dfrac{2BC^2}{2CB}=a\)
b: \(BC=8\sqrt{3}\cdot cos60+2\sqrt{2}\cdot cos45=4\sqrt{3}+2\left(cm\right)\)