C A D b a h x c B
Cosine Rule (proof) \begin{align} By\ trig\ ratios \quad x & = b\ cosA \\ \\ then\ by\ Pythagoras\ on\ & \triangle CDA \\ b^2 & = h^2+(b\ cosA)^2 \\ \implies\ h^2 & =b^2-(b\ cosA)^2 \\ \\ Also\ by\ Pythagoras\ & on\ \triangle CBD \\ a^2-(c- b\ cosA)^2 & = h^2 \\ \\ \therefore\ a^2-(c- b\ cosA)^2 & = b^2-(b\ cosA)^2 \\ \\ \implies\ a^2 = b^2+c^2- & 2bc\ cosA \\ \end{align}