\begin{align*} & \cot 70^\circ + 4\cos 70^\circ\\ =&\frac{\sin 20^\circ}{\cos 20^\circ} + 4\sin 20^\circ\\ =&\frac{\sin 20^\circ + 2\sin 40^\circ}{\cos 20^\circ}\\ =&\frac{2\sin 20^\circ\sin 30^\circ + 2\sin 40^\circ}{\cos 20^\circ}\\ =&\frac{\cos 10^\circ -\cos 50^\circ + 2\cos 50^\circ}{\cos 20^\circ}\\ =&\frac{\cos 10^\circ + \cos 50^\circ}{\cos 20^\circ}\\ =&\frac{2\cos 20^\circ \cos 20^\circ}{\cos 20^\circ}\\ =\sqrt{3} \end{align*}