The solution of the differential equation cos x sin y dx + sin x cos y dy = 0 is

The correct option/answer is sin x sin y = c.

Let us solve the given first-order differential equation step-by-step using the method of separation of variables.
The given differential equation is:
$$cosxsinydx+sinxcosydy=0$$

To separate the variables, we divide the entire equation by $sinxsiny$ (assuming $sinxsiny\ne 0$):
$$\frac{cosxsiny}{sinxsiny}dx+\frac{sinxcosy}{sinxsiny}dy=0$$

Simplifying the terms, we get:
$$\frac{cosx}{sinx}dx+\frac{cosy}{siny}dy=0$$

This can be written in terms of cotangent functions as:
$$cotxdx+cotydy=0$$

Now, we integrate both sides of the equation:
$$\int cotxdx+\int cotydy=C$$
where $C$ is an arbitrary constant of integration.

Recall the standard integration formula: $\int cotudu=ln|sinu|$. Applying this to our equation, we obtain:

where we have written the constant of integration as $ln\left|c\right|$ for convenience of simplification.

Using the logarithmic property $lnA+lnB=ln(A·B)$, we can combine the left side:
$$ln|sinxsiny|=ln\left|c\right|$$

Taking the exponential on both sides (exponentiating), we get:
$$sinxsiny=c$$

This matches the correct option.