Transformation

Relevant wiki: Euler's Formula

$\begin{aligned} X & = \frac {1-4\sin 10^\circ \sin 70^\circ}{2\sin 10^\circ} & \small \color{#3D99F6} \text{As }\sin \theta = \cos (90^\circ - \theta) \\ & = \frac {1-4\cos 80^\circ \cos 20^\circ}{2\cos 80^\circ} \\ & = \frac {1{\color{#3D99F6}-}4\cos \frac {4\pi}9 \color{#3D99F6}\cos \frac \pi 9}{2\cos \frac {4\pi}9} & \small \color{#3D99F6} \text{and }\cos (\pi - \theta) = - \cos \theta \\ & = \frac {1{\color{#3D99F6}+}4\cos \frac {4\pi}9 \color{#3D99F6}\cos \frac {8\pi}9}{2\cos \frac {4\pi}9} & \small \color{#3D99F6} \text{Let }\omega = e^{\frac {2\pi}9i} \text{ the 9th root of unity} \\ & = \frac {1+4\left(\frac {\omega^2 + \omega^{-2}}2\right)\left(\frac {\omega^4 + \omega^{-4}}2\right)}{2\left(\frac {\omega^2 + \omega^{-2}}2\right)} & \small \color{#3D99F6} \text{and by Euler's formula.} \\ & = \frac {1+(\omega^2 + \omega^{-2})(\omega^4 + \omega^{-4})}{\omega^2 + \omega^{-2}} \\ & = \frac {1+ {\color{#3D99F6}\omega^6} + \omega^{-2} + \omega^2 + \color{#3D99F6} \omega^{-6}}{\omega^2 + \omega^{-2}} & \small \color{#3D99F6} \text{Note that }\frac {\omega^6 + \omega^{-6}}2 = \cos \frac {2\pi}3 = - \frac 12 \\ & = \frac {1{\color{#3D99F6}-1} + \omega^{-2} + \omega^2}{\omega^2 + \omega^{-2}} \\ & = \boxed 1 \end{aligned}$