Considering L.H.S :
\[2\text{ }\left( bc\text{ }cos\text{ }A\text{ }+\text{ }ca\text{ }cos\text{ }B\text{ }+\text{ }ab\text{ }cos\text{ }C \right)\]
\[2ca\text{ }cos\text{ }B,\text{ }2ab\text{ }cos\text{ }C\text{ }and\text{ }2cb\text{ }cos\text{ }A,\]are present in L.H.S. which can be obtained from cosine formulae.
Using cosine formula:
\[Cos\text{ }A\text{ }=\text{ }({{b}^{2}}~+\text{ }{{c}^{2}}~\text{ }{{a}^{2}})/2bc\]
\[2bc\text{ }cos\text{ }A\text{ }=\text{ }({{b}^{2}}~+\text{ }{{c}^{2}}~\text{ }{{a}^{2}})\text{ }\ldots \text{ }\left( i \right)\]
Or,
\[Cos\text{ }B\text{ }=\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~\text{ }{{b}^{2}})/2ac\]
\[2ac\text{ }cos\text{ }B\text{ }=\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~\text{ }{{b}^{2}})\ldots \text{ }\left( ii \right)\]
Or,
\[Cos\text{ }C\text{ }=\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }{{c}^{2}})/2ab\]
\[2ab\text{ }cos\text{ }C\text{ }=\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }{{c}^{2}})\text{ }\ldots \text{ }\left( iii \right)\]
Adding equation (i), (ii) and (ii) we get,
\[2bc\text{ }cos\text{ }A\text{ }+\text{ }2ac\text{ }cos\text{ }B\text{ }+\text{ }2ab\text{ }cos\text{ }C\]
\[=\text{ }({{b}^{2}}~+\text{ }{{c}^{2}}~\text{ }{{a}^{2}})\text{ }+\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~\text{ }{{b}^{2}})\text{ }+\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }{{c}^{2}})\]
Simplifying,
\[=\text{ }{{c}^{2}}~+\text{ }{{b}^{2}}~+\text{ }{{a}^{2}}\]
So,
\[2\text{ }\left( bc\text{ }cos\text{ }A\text{ }+\text{ }ac\text{ }cos\text{ }B\text{ }+\text{ }ab\text{ }cos\text{ }C \right)\]
\[=\text{ }{{a}^{2}}~+\text{ }{{b}^{2}}~+\text{ }{{c}^{2}}\]
Hence proved.