Parallelogram trigonometry
For part A, try counting the area of the parallelogram in two different ways, as suggested by Jim Belk.
For part B, notice that your diagram has $n$ and $m$ reversed, since $m<n$. In particular, $\alpha$ should be opposite $m$, not $n$. The modified version of your formula for $m$ is then $$m^2 = a^2+b^2 -2ab\cos\alpha.$$
Try combining this with your formula $$4a^2 = m^2 + n^2 -2mn\cos\beta$$ and use the result from part A.