Considering L.H.S :
Getting terms c – b cos A and b – c cos A from projection formula
Using projection formula,
\[c\text{ }=\text{ }a\text{ }cos\text{ }B\text{ }+\text{ }b\text{ }cos\text{ }A\]
\[c\text{ }\text{ }b\text{ }cos\text{ }A\text{ }=\text{ }a\text{ }cos\text{ }B\text{ }\ldots .\text{ }\left( i \right)\]
And,
\[b\text{ }=\text{ }c\text{ }cos\text{ }A\text{ }+\text{ }a\text{ }cos\text{ }C\]
\[b\text{ }\text{ }c\text{ }cos\text{ }A\text{ }=\text{ }a\text{ }cos\text{ }C\text{ }\ldots .\text{ }\left( ii \right)\]
On dividing equation (i) by (ii),
\[=\text{ }RHS\]
Hence Proved.