What is the solution of the differential equation
` -cosec^(2)(x+y)dy=dx` ?

Options

A. y-c=sin (x+y)
B. x-c= sin (x+y)
C. y-c = tan (x +y)
D. none of the above

Explanation

To solve the differential equation \(-\csc^2(x+y) \, dy = dx\), we can follow these steps:

### Step 1: Rearranging the equation
We start by rewriting the equation in a more manageable form:
\[ \csc^2(x+y) \, dy = -dx \]

### Step 2: Separating variables
Now, we can separate the variables \(y\) and \(x\):
\[ dy = -\csc^2(x+y) \, dx \]

### Step 3: Integrating both sides
Next, we integrate both sides. The left side involves \(y\) and the right side involves \(x\):
\[ \int dy = -\int \csc^2(x+y) \, dx \]

The integral of \(dy\) is simply \(y\). For the right side, we can use the substitution \(u = x + y\), which gives us \(du = dx + dy\). Thus, we can rewrite the integral:
\[ \int \csc^2(u) \, du = -\cot(u) + C \]
So we have:
\[ y = -\cot(x+y) + C \]

### Step 4: Solving for \(y\)
Now we need to express \(y\) explicitly. Rearranging gives:
\[ y + \cot(x+y) = C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y + \cot(x+y) = C \]

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