1. **Problem statement:** Solve the differential equation $$\cos(x) \cos(y) \, dx + \sin(x) \sin(y) \, dy = 0.$$\n\n2. **Rewrite and rearrange:** We want to express \( \frac{dy}{dx} \) explicitly if possible.\nDivide both sides by \(dx\) (assuming \(dx \neq 0\)):\n$$\cos(x) \cos(y) + \sin(x) \sin(y) \frac{dy}{dx} = 0.$$\nHence,\n$$\frac{dy}{dx} = - \frac{\cos(x) \cos(y)}{\sin(x) \sin(y)}.$$\n\n3. **Simplify the derivative:** Write \( \frac{dy}{dx} = - \frac{\cos(x)}{\sin(x)} \cdot \frac{\cos(y)}{\sin(y)} = - \cot(x) \cot(y).$$\n\n4. **Separate variables:** The equation becomes\n$$\frac{dy}{dx} = - \cot(x) \cot(y) \Rightarrow \frac{dy}{\cot(y)} = - \cot(x) \, dx.$$\nSince \(\cot(y) = \frac{\cos(y)}{\sin(y)}\) and \(\frac{1}{\cot(y)} = \tan(y)\), rewrite as\n$$\frac{dy}{\cot(y)} = \tan(y) \, dy.$$\nSo we have\n$$\tan(y) \, dy = - \cot(x) \, dx.$$\n\n5. **Integrate both sides:**\n$$\int \tan(y) \, dy = - \int \cot(x) \, dx.$$\nRecall:\n$$\int \tan(y) \, dy = - \ln|\cos(y)| + C_1,$$\n$$\int \cot(x) \, dx = \ln|\sin(x)| + C_2.$$\nHence,\n$$- \ln|\cos(y)| = - \ln|\sin(x)| + C,$$\nwhere \(C = C_2 - C_1\).\n\n6. **Simplify the expression:** Multiply both sides by -1:\n$$\ln|\cos(y)| = \ln|\sin(x)| - C.$$\nExponentiate both sides:\n$$|\cos(y)| = e^{-C} |\sin(x)|.$$\nLet \(K = e^{-C} > 0\) be an arbitrary positive constant. So,\n$$|\cos(y)| = K |\sin(x)|.$$\n\n7. **Final implicit general solution:**\n$$\cos(y) = C \sin(x),$$\nwhere \(C\) is an arbitrary constant (absorbing signs).\n\nThis implicit relation defines the general solution of the differential equation.