` cosx cos ydy - sin x sin y dx =0`

Answer

To solve the differential equation \( \cos x \cos y \, dy - \sin x \sin y \, dx = 0 \), we will follow a systematic approach.

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate the differentials:
\[ \cos x \cos y \, dy = \sin x \sin y \, dx \]
Now, we can separate the variables:
\[ \frac{dy}{\sin y} = \frac{\sin x}{\cos x} \, dx \]

### Step 2: Simplifying the Right Side
We know that \( \frac{\sin x}{\cos x} = \tan x \), so we can rewrite the equation as:
\[ \frac{dy}{\sin y} = \tan x \, dx \]

### Step 3: Integrating Both Sides
Now we will integrate both sides:
\[ \int \frac{dy}{\sin y} = \int \tan x \, dx \]
The left side integrates to:
\[ \int \frac{dy}{\sin y} = \log |\tan(\frac{y}{2})| + C_1 \]
And the right side integrates to:
\[ \int \tan x \, dx = -\log |\cos x| + C_2 \]

### Step 4: Combining the Results
Combining the results of the integrals, we have:
\[ \log |\tan(\frac{y}{2})| = -\log |\cos x| + C \]
where \( C = C_2 - C_1 \).

### Step 5: Exponentiating Both Sides
Exponentiating both sides gives us:
\[ |\tan(\frac{y}{2})| = \frac{C}{|\cos x|} \]

### Step 6: Final Solution
Thus, we can express the solution as:
\[ \tan(\frac{y}{2}) \cos x = C \]
where \( C \) is a constant.