1. We are given the differential equation: $$\cos(x)\cos(y)\,dx + \sin(x)\sin(y)\,dy = 0$$ 2. To find the general solution, first rewrite in the form: $$M(x,y)\,dx + N(x,y)\,dy = 0$$ where $$M = \cos(x)\cos(y)$$ and $$N = \sin(x)\sin(y)$$. 3. Check if the equation is exact by computing partial derivatives: $$\frac{\partial M}{\partial y} = -\cos(x)\sin(y)$$ $$\frac{\partial N}{\partial x} = \cos(x)\sin(y)$$ Since $$\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$$, the equation is not exact. 4. Try the substitution $$z = \frac{\tan(x)}{\tan(y)}$$ or manipulate to separate variables. Rewrite the equation: $$\cos(x)\cos(y)\,dx = -\sin(x)\sin(y)\,dy$$ Divide both sides by $$\cos(x)\cos(y)$$: $$dx = -\tan(x)\tan(y)\,dy$$ or $$\frac{dx}{dy} = -\tan(x)\tan(y)$$ 5. Rewrite as: $$\frac{dx}{dy} = -\tan(x)\tan(y)$$ Separate variables: $$\frac{dx}{\tan(x)} = -\tan(y)\,dy$$ 6. Recall $$\frac{1}{\tan(x)} = \cot(x)$$, so: $$\cot(x)\,dx = -\tan(y)\,dy$$ 7. Integrate both sides: $$\int \cot(x)\,dx = - \int \tan(y)\,dy$$ 8. Recall integrals: $$\int \cot(x)\,dx = \ln|\sin(x)| + C_1$$ $$\int \tan(y)\,dy = -\ln|\cos(y)| + C_2$$ 9. So: $$\ln|\sin(x)| = -(-\ln|\cos(y)|) + C$$ or $$\ln|\sin(x)| = \ln|\cos(y)| + C$$ 10. Exponentiate both sides: $$|\sin(x)| = K|\cos(y)|$$ where $$K = e^C$$ 11. Therefore, the general implicit solution is: $$\sin(x) = C \cos(y)$$ where $$C$$ is an arbitrary constant. Final answer: $$\boxed{\sin(x) = C \cos(y)}$$