To solve the differential equation $Y'(x) = \frac{\sin(x)}{\cos(Y(x))}$, let’s rewrite it in a separable form:

1. Start by rewriting $Y'(x) = \frac{dY}{dx}$ as: $\frac{dY}{dx} = \frac{\sin(x)}{\cos(Y)}$

2. Separate variables, moving all $Y$-related terms to one side and $x$-related terms to the other: $\cos(Y) \, dY = \sin(x) \, dx$

3. Now, integrate both sides: $\int \cos(Y) \, dY = \int \sin(x) \, dx$

4. Integrating each side:

   • The integral of $\cos(Y)$ with respect to $Y$ is $\sin(Y)$.
   • The integral of $\sin(x)$ with respect to $x$ is $-\cos(x)$.

   Thus, we get: $\sin(Y) = -\cos(x) + C$ where $C$ is the constant of integration.

5. Solving for $Y$, we obtain the implicit solution: $Y = \arcsin(-\cos(x) + C)$

This is the general solution of the differential equation.

Would you like more details on this solution or have any questions?

Here are five related questions to extend your understanding:

1. How do we determine $C$ if an initial condition is given?
2. What is the domain of the solution based on the properties of $\arcsin$?
3. How can this differential equation be solved using numerical methods?
4. How would the solution change if $Y'(x)$ had a different trigonometric expression?
5. What are some applications of such separable differential equations in real-world contexts?

Tip: When solving separable differential equations, always check if both sides are easily integrable before separating.