Find the general solution of the following :
`(1+cos x)(dy)/(dx)= (1-cos x)`

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x} \]

### Step 2: Simplifying the Right Side
Next, we simplify the right side of the equation:
\[ \frac{1 - \cos x}{1 + \cos x} \]
We can use the identity \(1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right)\) and \(1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right)\):
\[ \frac{1 - \cos x}{1 + \cos x} = \frac{2 \sin^2\left(\frac{x}{2}\right)}{2 \cos^2\left(\frac{x}{2}\right)} = \tan^2\left(\frac{x}{2}\right) \]

### Step 3: Integrating Both Sides
Now, we can integrate both sides:
\[ dy = \tan^2\left(\frac{x}{2}\right) dx \]
Integrating the left side gives us \(y\), and we need to integrate the right side:
\[ \int dy = \int \tan^2\left(\frac{x}{2}\right) dx \]

Using the identity \(\tan^2 A = \sec^2 A - 1\), we can rewrite the integral:
\[ \int \tan^2\left(\frac{x}{2}\right) dx = \int \left(\sec^2\left(\frac{x}{2}\right) - 1\right) dx \]

### Step 4: Performing the Integration
Now we perform the integration:
\[ \int \sec^2\left(\frac{x}{2}\right) dx - \int dx \]
The integral of \(\sec^2\left(\frac{x}{2}\right)\) is \(2 \tan\left(\frac{x}{2}\right)\) (using the substitution \(u = \frac{x}{2}\)):
\[ = 2 \tan\left(\frac{x}{2}\right) - x + C \]

### Step 5: Final Solution
Thus, the general solution of the differential equation is:
\[ y = 2 \tan\left(\frac{x}{2}\right) - x + C \]