Solve the differential equation: cosec³ x dy - cosec y dx = 0

$\operatorname{cosec}^3 x\,dy = \operatorname{cosec} y\,dx$ Separating variables: $\sin y\,dy = \sin^3 x\,dx$ $\sin^3 x = \sin x(1-\cos^2 x)$, so $\int \sin^3 x\,dx = -\cos x + \frac{\cos^3 x}{3}$ Integrating: $-\cos y = -\cos x + \frac{\cos^3 x}{3} + c$ Hence $\cos y = \cos x - \frac{\cos^3 x}{3} + C$.