The curves $C_1$ and $C_2$ have equations $y=1+4 \cos 2x$ and $y=2 {\cos }^{2}x-4 \sin 2x.$

Show that the $x$-coordinates of the points where $C_1$ and $C_2$ intersect satisfy the equation $3 \cos 2x+4 \sin 2x=0.$

The curves $C_1$ and $C_2$ have equations $y=1+4 \cos 2x$ and $y=2 {\cos }^{2}x-4 \sin 2x.$ Express $3 \cos 2\mathrm{x}+4 \sin 2\mathrm{x}$ in the form $R \sin (2x+\alpha ),$ where $R>0$ and $0°<\alpha <90°$ and  give the exact value of $R$ and the value of $\alpha$ correct to $2$ decimal places.

The curves $C_1$ and $C_2$ have equations $y=1+4 \cos 2x$ and $y=2 {\cos }^{2}x-4 \sin 2x.$ Find all the roots of the equation $3 \cos 2x+4 \sin 2x=0$ for $0°<x<180°.$