A laser tracking system uses a sensor where the angle $x$ (in radians) of the beam relative to a horizontal axis must satisfy the equation:

$$5{\sec }^{2}x+7\tan x=5$$

Determine all solutions for $x$ in the interval $-\pi <x\le 0$, giving your answers to 3 significant figures where appropriate.

In the study of resonance in a harmonic oscillator, an engineer encounters the following expression involving phase angles. Help them prove the identity:

$$\frac{\sin 7\beta }{\sin 2\beta }-\frac{\cos 7\beta }{\cos 2\beta }\equiv \frac{\sin 5\beta }{\sin 2\beta \cos 2\beta }$$