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

Options

A. `(sinx)/(siny) = c`
B. `cos x + cos y = c`
C. `sin x + sin y = c`
D. `sin x. sin y=c`

Explanation

To solve the differential equation \( \cos x \sin y \, dx + \sin x \cos y \, dy = 0 \), we can follow these steps:

### Step 1: Rearrange the Equation
We start by rearranging the given equation:
\[ \cos x \sin y \, dx + \sin x \cos y \, dy = 0 \]
This can be rewritten as:
\[ \cos x \sin y \, dx = -\sin x \cos y \, dy \]

### Step 2: Separate Variables
Next, we separate the variables \( x \) and \( y \):
\[ \frac{dx}{\sin x} = -\frac{\cos y}{\sin y} \, dy \]
This simplifies to:
\[ \frac{dx}{\sin x} = -\cot y \, dy \]

### Step 3: Integrate Both Sides
Now, we integrate both sides:
\[ \int \frac{dx}{\sin x} = -\int \cot y \, dy \]
The integral of \( \frac{1}{\sin x} \) is \( \log |\tan(\frac{x}{2})| \) and the integral of \( \cot y \) is \( \log |\sin y| \). Thus, we have:
\[ \log |\tan(\frac{x}{2})| = -\log |\sin y| + C \]

### Step 4: Simplify the Equation
We can rewrite the equation as:
\[ \log |\tan(\frac{x}{2})| + \log |\sin y| = C \]
Using properties of logarithms, this can be combined:
\[ \log |\tan(\frac{x}{2}) \sin y| = C \]

### Step 5: Exponentiate to Remove Logarithm
Exponentiating both sides gives:
\[ |\tan(\frac{x}{2}) \sin y| = e^C \]
Let \( K = e^C \), then:
\[ \tan(\frac{x}{2}) \sin y = K \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \tan(\frac{x}{2}) \sin y = C \]
where \( C \) is a constant.

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