Prove a projection formula in two ways

The angle between two unit vectors $\mathbf a$ and $\mathbf b$ is $\theta$.

By expanding the scalar product, show that $(3\mathbf a-\mathbf b)\cdot(\mathbf a+3\mathbf b)=8\cos\theta$.

Given that $\theta=60^\circ$, show that the length of projection of $(3\mathbf a-\mathbf b)$ onto $(\mathbf a+3\mathbf b)$ is $\dfrac4{\sqrt{13}}$.