`sec x dy +cosec y dx=0`

Options

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

Answer

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

### Step 1: Rearranging the equation
We start with the given equation:
\[ \sec x \, dy + \csc y \, dx = 0 \]
We can rearrange this to isolate \( dy \) and \( dx \):
\[ \sec x \, dy = -\csc y \, dx \]
Dividing both sides by \( \sec x \) and \( \csc y \):
\[ \frac{dy}{\csc y} = -\frac{dx}{\sec x} \]

### Step 2: Simplifying the fractions
Recall that \( \csc y = \frac{1}{\sin y} \) and \( \sec x = \frac{1}{\cos x} \). Thus, we can rewrite the equation:
\[ \sin y \, dy = -\cos x \, dx \]

### Step 3: Integrating both sides
Now we integrate both sides:
\[ \int \sin y \, dy = -\int \cos x \, dx \]
The left side integrates to:
\[ -\cos y \]
And the right side integrates to:
\[ -\sin x + C \]
Thus, we have:
\[ -\cos y = -\sin x + C \]

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

### Final Solution
The final solution to the differential equation is:
\[ \sin x - \cos y = C \]