In a stability analysis for a rotating pendulum, the equilibrium angle α\alphaα in the interval 0<α≤π0 < \alpha \le \pi0<α≤π satisfies the equation
2cosec2α−5cotα=52\text{cosec}^2 \alpha - 5\cot \alpha = 52cosec2α−5cotα=5
Determine all possible values for α\alphaα, giving your answers to 3 significant figures.
Prove the following trigonometric identity for all valid values of θ\thetaθ:
sin8θsin3θ−cos8θcos3θ≡sin5θsin3θcos3θ\frac{\sin 8\theta}{\sin 3\theta} - \frac{\cos 8\theta}{\cos 3\theta} \equiv \frac{\sin 5\theta}{\sin 3\theta \cos 3\theta}sin3θsin8θ−cos3θcos8θ≡sin3θcos3θsin5θ