Solve the following differential equation: `(dy)/(dx)=(1-cos2y)/(1+cos2y)`

Explanation

To solve the differential equation \(\frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y}\), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y} \]
Since the right-hand side is a function of \(y\) only, we can rewrite the equation as:
\[ dx = \frac{1 + \cos 2y}{1 - \cos 2y} dy \]

### Step 2: Integrate Both Sides
Now, we integrate both sides:
\[ \int dx = \int \frac{1 + \cos 2y}{1 - \cos 2y} dy \]
The left side integrates to:
\[ x + C_1 \]
where \(C_1\) is a constant of integration.

### Step 3: Simplify the Right Side
Next, we simplify the right-hand side. We can use the trigonometric identities:
\[ \cos 2y = 2\cos^2 y - 1 \quad \text{and} \quad 1 - \cos 2y = 2\sin^2 y \]
Thus,
\[ 1 + \cos 2y = 2\cos^2 y \]
So, we can rewrite the integral:
\[ \int \frac{2\cos^2 y}{2\sin^2 y} dy = \int \cot^2 y \, dy \]

### Step 4: Integrate \(\cot^2 y\)
We know that:
\[ \cot^2 y = \csc^2 y - 1 \]
Thus, we can split the integral:
\[ \int \cot^2 y \, dy = \int (\csc^2 y - 1) \, dy = \int \csc^2 y \, dy - \int 1 \, dy \]
The integrals yield:
\[ -\cot y - y + C_2 \]
where \(C_2\) is another constant of integration.

### Step 5: Combine Results
Now, we combine both sides:
\[ x + C_1 = -\cot y - y + C_2 \]
Rearranging gives:
\[ x + \cot y + y = C \]
where \(C = C_2 - C_1\) is a constant.

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