The solution of `cos(x+y)dy=dx` is

Options

A. `y=tan((x+y)/(2))+C`
B. `y=cos^(-1)((y)/(x))`
C. `y=x sec ((y)/(x))+C`
D. none of these

Explanation

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos(x+y) \, dy = dx \]
We can rearrange this to express \( dy \) in terms of \( dx \):
\[ dy = \frac{dx}{\cos(x+y)} \]

### Step 2: Substitute \( v = x + y \)
Let’s make a substitution to simplify the equation. Set:
\[ v = x + y \]
Then, differentiating both sides with respect to \( x \) gives:
\[ \frac{dv}{dx} = 1 + \frac{dy}{dx} \]
Thus, we can express \( \frac{dy}{dx} \) as:
\[ \frac{dy}{dx} = \frac{dv}{dx} - 1 \]

### Step 3: Substitute into the equation
Now, substitute \( \frac{dy}{dx} \) in the rearranged equation:
\[ \frac{dv}{dx} - 1 = \frac{1}{\cos(v)} \]
This simplifies to:
\[ \frac{dv}{dx} = \frac{1}{\cos(v)} + 1 \]

### Step 4: Combine terms
We can combine the terms on the right-hand side:
\[ \frac{dv}{dx} = \frac{1 + \cos(v)}{\cos(v)} \]

### Step 5: Separate variables
Now, we separate the variables:
\[ \frac{\cos(v)}{1 + \cos(v)} \, dv = dx \]

### Step 6: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{\cos(v)}{1 + \cos(v)} \, dv = \int dx \]
The left-hand side can be simplified:
\[ \int \left( 1 - \frac{1}{1 + \cos(v)} \right) dv = \int dx \]
This gives us:
\[ \int dv - \int \frac{1}{1 + \cos(v)} \, dv = x + C \]

### Step 7: Solve the integrals
The first integral is straightforward:
\[ v - \int \frac{1}{1 + \cos(v)} \, dv = x + C \]
Using the half-angle identity, we can express \( 1 + \cos(v) \) as:
\[ 1 + \cos(v) = 2 \cos^2\left(\frac{v}{2}\right) \]
Thus:
\[ \int \frac{1}{1 + \cos(v)} \, dv = \int \frac{1}{2 \cos^2\left(\frac{v}{2}\right)} \, dv = \frac{1}{2} \tan\left(\frac{v}{2}\right) + C \]
Putting this back, we get:
\[ v - \frac{1}{2} \tan\left(\frac{v}{2}\right) = x + C \]

### Step 8: Substitute back for \( v \)
Since \( v = x + y \), we substitute back:
\[ x + y - \frac{1}{2} \tan\left(\frac{x + y}{2}\right) = x + C \]
This simplifies to:
\[ y - \frac{1}{2} \tan\left(\frac{x + y}{2}\right) = C \]

### Final Solution
Thus, the general solution of the differential equation is:
\[ y = \frac{1}{2} \tan\left(\frac{x + y}{2}\right) + C \]