\begin{align*} &\cot 70^\circ + 4\cos 70^\circ\\ &=\tan 20^\circ + 4\sin 20^\circ\\ &=\frac{\sin 20^\circ}{\cos 20^\circ} + 4\sin 20^\circ\\ &=\frac{1}{\cos 20^\circ}\cdot(\sin 20^\circ + 4\sin 20^\circ\cos 20^\circ)\\ &=\frac{1}{\cos 20^\circ}\cdot(\sin 20^\circ + 2\sin 40^\circ)\\ &=\frac{1}{\cos 20^\circ}\cdot((\sin 20^\circ + \sin 40^\circ)+\sin 40^\circ)\\ &=\frac{1}{\cos 20^\circ}\cdot(2\sin 30^\circ \cos 10^\circ + \sin 40^\circ )\\ &=\frac{1}{\cos 20^\circ}\cdot(\cos 10^\circ + \sin 40^\circ)\\ &=\frac{1}{\cos 20^\circ}\cdot(\sin 80^\circ + \sin 40^\circ)\\ &=\frac{1}{\cos 20^\circ}\cdot 2 \cdot \sin 60^\circ \cos 20^\circ\\ &= 2\sin 60^\circ\\ &=\sqrt{3} \end{align*}