1. **State the problem:** Solve the differential equation $$\csc y \, dx + \sec^2 x \, dy = 0$$ by separation of variables. 2. **Rewrite the equation:** We have $$\csc y \, dx + \sec^2 x \, dy = 0 \implies \csc y \, dx = -\sec^2 x \, dy$$ 3. **Separate variables:** Rearranging, $$\frac{dx}{\sec^2 x} = -\frac{dy}{\csc y}$$ Recall that $$\frac{1}{\sec^2 x} = \cos^2 x$$ and $$\frac{1}{\csc y} = \sin y$$, so $$\cos^2 x \, dx = -\sin y \, dy$$ 4. **Integrate both sides:** $$\int \cos^2 x \, dx = -\int \sin y \, dy$$ 5. **Integrate left side:** Use the identity $$\cos^2 x = \frac{1 + \cos 2x}{2}$$, $$\int \cos^2 x \, dx = \int \frac{1 + \cos 2x}{2} \, dx = \frac{1}{2} \int 1 \, dx + \frac{1}{2} \int \cos 2x \, dx = \frac{x}{2} + \frac{1}{4} \sin 2x + C_1$$ 6. **Integrate right side:** $$-\int \sin y \, dy = -(-\cos y) + C_2 = \cos y + C_2$$ 7. **Combine constants:** Let $$C = C_2 - C_1$$, then $$\frac{x}{2} + \frac{1}{4} \sin 2x = \cos y + C$$ 8. **Final implicit solution:** $$\frac{x}{2} + \frac{1}{4} \sin 2x - \cos y = C$$ This is the implicit general solution to the differential equation.