<s>[INST] Simplifying $\tan100^{\circ}+4\sin100^{\circ}$ The answer is $-\sqrt3$. 
I was wondering if this is just a coincidence?

Also, is there a relation between $$\tan(100^{\circ}+20^{\circ})=\frac{\tan100^{\circ}+\tan20^{\circ}}{1-\tan100^{\circ}.\tan20^{\circ}}=-\sqrt3$$ and the given expression? Or is there a more elegant method of solving the question? 
 [/INST] One has $$\tan 100^\circ + 4\sin 100^\circ = \frac{\sin 100^\circ + 2\sin 200^\circ}{\cos 100^\circ} = \frac{\sin 100^\circ - 2\sin 20^\circ}{\cos 100^\circ} = \frac{2\cos 60^\circ\sin 40^\circ - \sin 20^\circ}{\cos 100^\circ} = \frac{\sin 40^\circ - \sin 20^\circ}{\cos 100^\circ} = \frac{2\cos 30^\circ\sin 10^\circ }{\cos 100^\circ} = -\sqrt{3}.$$
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