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

Answer

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

### Step 1: Rearrange the equation
We start by rewriting the equation in a more manageable form:

\[ dy = \frac{1 - \cos x}{1 + \cos x} dx \]

### Step 2: Integrate both sides
Next, we will integrate both sides. The left side integrates to \(y\), and we need to integrate the right side:

\[ y = \int \frac{1 - \cos x}{1 + \cos x} dx \]

### Step 3: Simplify the integrand
To simplify \(\frac{1 - \cos x}{1 + \cos x}\), 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 4: Integrate \(\tan^2\left(\frac{x}{2}\right)\)
Now we can integrate:

\[ y = \int \tan^2\left(\frac{x}{2}\right) dx \]

Using the identity \(\tan^2 A = \sec^2 A - 1\), we rewrite the integral:

\[ y = \int \left(\sec^2\left(\frac{x}{2}\right) - 1\right) dx \]

### Step 5: Integrate term by term
Now we integrate term by term:

1. The integral of \(\sec^2\left(\frac{x}{2}\right)\) is \(2 \tan\left(\frac{x}{2}\right)\).
2. The integral of \(-1\) is \(-x\).

Thus, we have:

\[ y = 2 \tan\left(\frac{x}{2}\right) - x + C \]

where \(C\) is the constant of integration.

### Final Solution
The general solution of the differential equation is:

\[ y = 2 \tan\left(\frac{x}{2}\right) - x + C \]

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