Step 1: Separate the variables.

The given differential equation is:

$$\sin x \cos y \, dx + \cos x \sin y \, dy = 0$$

Rearranging the terms to separate variables $x$ and $y$:

$$\cos x \sin y \, dy = - \sin x \cos y \, dx$$

Dividing both sides by $\cos x \cos y$ (assuming $\cos x \neq 0, \cos y \neq 0$):

$$\frac{\sin y}{\cos y} \, dy = - \frac{\sin x}{\cos x} \, dx$$ $$\tan y \, dy = - \tan x \, dx$$

Step 2: Integrate both sides.

Integrating the left side with respect to $y$ and the right side with respect to $x$:

$$\int \tan y \, dy = - \int \tan x \, dx$$

Using the standard integral formula $\int \tan \theta \, d\theta = \log |\sec \theta| + C$:

$$\log |\sec y| = - \log |\sec x| + \log C$$ $$\log |\sec y| + \log |\sec x| = \log C$$ $$\log |\sec x \sec y| = \log C$$

This simplifies to:

$$\sec x \sec y = C$$

Or:

$$\cos x \cos y = \frac{1}{C} = C_1 \text{ (new constant)}$$

Step 3: Find the particular solution.

Substitute the point $\left(0, \frac{\pi}{4}\right)$ into the equation $\cos x \cos y = C_1$:

$$\cos(0) \cos\left(\frac{\pi}{4}\right) = C_1$$

Since $\cos(0) = 1$ and $\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$:

$$1 \times \frac{1}{\sqrt{2}} = C_1$$ $$C_1 = \frac{1}{\sqrt{2}}$$

Step 4: Write the final equation.

Substituting the value of $C_1$ back into the equation:

$$\cos x \cos y = \frac{1}{\sqrt{2}}$$

This is the equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$.