## Solving the Differential Equation: (tan(x) − sin(x) sin(y)) dx + cos(x) cos(y) dy = 0

This differential equation is of the form **M(x, y) dx + N(x, y) dy = 0**, where:

**M(x, y) = tan(x) − sin(x) sin(y)****N(x, y) = cos(x) cos(y)**

To solve this, we can check if it's an **exact differential equation**. This means that the partial derivative of M with respect to y should be equal to the partial derivative of N with respect to x. Let's check:

**∂M/∂y = -sin(x) cos(y)****∂N/∂x = -sin(x) cos(y)**

Since **∂M/∂y = ∂N/∂x**, the equation is **exact**. This means there exists a function **F(x, y)** such that:

**∂F/∂x = M(x, y)****∂F/∂y = N(x, y)**

To find F(x, y), we integrate M(x, y) with respect to x, treating y as a constant:

**F(x, y) = ∫ (tan(x) − sin(x) sin(y)) dx = ln|sec(x)| + cos(x) sin(y) + g(y)**

Here, g(y) is an arbitrary function of y. Now, we differentiate F(x, y) with respect to y and compare it to N(x, y):

**∂F/∂y = cos(x) cos(y) + g'(y)**

Comparing this with N(x, y) = cos(x) cos(y), we find that g'(y) = 0. Integrating this gives us:

**g(y) = C**, where C is an arbitrary constant.

Therefore, the general solution to the differential equation is:

**F(x, y) = ln|sec(x)| + cos(x) sin(y) + C = 0**

This solution represents a family of curves that are implicitly defined by the equation.