The solution of the differential equation ` sec x dy - "cosec " y dx = 0 ` is

Options

A. `cos x + sin y= C`
B. ` sin x + cos y = C`
C. `sin y - cos x = C`
D. ` cos y - sin x = C`

Explanation

To solve the differential equation \( \sec x \, dy - \csc y \, dx = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ \sec x \, dy - \csc y \, dx = 0 \]
Rearranging this gives:
\[ \sec x \, dy = \csc y \, dx \]

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

### Step 3: Integrating Both Sides
Now, we integrate both sides:
\[ \int \sin y \, dy = \int \cos x \, dx \]
The integrals yield:
\[ -\cos y = \sin x + C \]
where \( C \) is the constant of integration.

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ \cos y = -\sin x - C \]

### Step 5: Final Form
To express this in a more standard form, we can write:
\[ \sin x + \cos y = -C \]
Letting \( -C \) be a new constant \( C' \), we can write:
\[ \sin x + \cos y = C' \]

### Conclusion
Thus, the solution to the differential equation is:
\[ \sin x + \cos y = C \]