Using the diagram below, prove the cosine rule.

Use Pythagoras to find a relationship between $c,h$ and $x.$

Do similar to find a relationship between $a,h$ and $(b-x).$

In the previous hint, we found two equations using Pythagoras. Eliminate $h$ from these equations (either by substituting $h^2$ from one equation into the other, or by subtracting the equations).

Notice that $\cos A=\frac{x}{c}.$ Use this to eliminate $x$ from the equation found in the previous hint.