$\cot 70^\circ +4\cos70^\circ=$
$=\frac{\cos 70^\circ}{\sin 70^\circ} +4\cos70^\circ=$
$=\frac{\cos 70^\circ + 4\sin 70^\circ \cos 70^\circ}{\sin 70^\circ }$
$=\frac{\cos 70^\circ +2(2\sin 70^\circ \cos 70^\circ)}{\sin 70^\circ }$
$=\frac{\cos 70^\circ +2\sin 140^\circ}{\sin 70^\circ }$
$=\frac{\sin 20^\circ +2\sin 40^\circ}{\sin 70^\circ }$
$=\frac{(\sin 20^\circ +\sin 40^\circ) + \sin 40^\circ}{\sin 70^\circ }$
$=\frac{(2 \sin 30^\circ \cos 10^\circ) + \sin 40^\circ}{\sin 70^\circ }$ using sin A + sin B formula
$=\frac{\cos 10^\circ + \sin 40^\circ}{\sin 70^\circ }$
$=\frac{\sin 80^\circ + \sin 40^\circ}{\sin 70^\circ }$
$=\frac{2 * \sin 60^\circ \cos 20^\circ}{\sin 70^\circ }$
$=\frac{\sqrt{3} \sin 70^\circ}{\sin 70^\circ }$
$=\sqrt{3}$
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