In a study of signal interference, the intensity of a resultant wave is modeled by a function containing trigonometric ratios.
Prove that
cos2ϕsinϕ+sin2ϕcosϕ≡cscϕ,ϕ≠nπ2,n∈Z \frac{\cos 2\phi}{\sin \phi} + \frac{\sin 2\phi}{\cos \phi} \equiv \csc \phi, \quad \phi \neq \frac{n\pi}{2}, n \in \mathbb{Z} sinϕcos2ϕ+cosϕsin2ϕ≡cscϕ,ϕ=2nπ,n∈ZHence solve, for 0≤θ<2π0 \le \theta < 2\pi0≤θ<2π,
2(cos4θsin2θ+sin4θcos2θ)+3cot22θ=5 2 \left( \frac{\cos 4\theta}{\sin 2\theta} + \frac{\sin 4\theta}{\cos 2\theta} \right) + 3\cot^2 2\theta = 5 2(sin2θcos4θ+cos2θsin4θ)+3cot22θ=5giving your answers in radians to 3 significant figures where appropriate.