Trigonometry Formulae - Part 10

$\begin{aligned} X & = \sin 50^\circ (1+\sqrt 3 \tan 10^\circ) \\ & = \sin (60^\circ - 10^\circ) \times \frac {\cos 10^\circ +\sqrt 3 \sin 10^\circ}{\cos 10^\circ} \\ & = \left(\frac {\sqrt 3}2 \cos 10^\circ - \frac 12 \sin 10^\circ \right)\frac {\cos 10^\circ +\sqrt 3 \sin 10^\circ}{\cos 10^\circ} \\ & = \frac {\sqrt 3\cos^2 10^\circ + 2\sin 10^\circ \cos 10^\circ - \sqrt 3 \sin^2 10^\circ}{2\cos 10^\circ} \\ & = \frac {\sqrt 3\cos 20^\circ + \sin 20^\circ}{2\cos 10^\circ} \\ & = \frac {\sin 60^\circ \cos 20^\circ + \cos 60^\circ \sin 20^\circ}{\cos 10^\circ} \\ & = \frac {\sin 80^\circ}{\cos 10^\circ} = \frac {\cos 10^\circ}{\cos 10^\circ} = \boxed 1 \end{aligned}$