If `y=x tan ""x/2`, then value of `(1+cos x) (dy)/(dx)-sin x` is

Options

A. `-x`
B. `x^(2)`
C. x
D. None of these

Answer

To solve the problem, we need to find the value of the expression \((1 + \cos x) \frac{dy}{dx} - \sin x\) where \(y = x \tan\left(\frac{x}{2}\right)\).

### Step 1: Find \(\frac{dy}{dx}\)

Given:
\[ y = x \tan\left(\frac{x}{2}\right) \]

We will use the product rule and chain rule to differentiate \(y\).

Using the product rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(x) \cdot \tan\left(\frac{x}{2}\right) + x \cdot \frac{d}{dx}\left(\tan\left(\frac{x}{2}\right)\right) \]

The derivative of \(x\) is \(1\), and now we need to find \(\frac{d}{dx}\left(\tan\left(\frac{x}{2}\right)\right)\) using the chain rule:
\[ \frac{d}{dx}\left(\tan\left(\frac{x}{2}\right)\right) = \sec^2\left(\frac{x}{2}\right) \cdot \frac{1}{2} \]

So, we have:
\[ \frac{dy}{dx} = \tan\left(\frac{x}{2}\right) + x \cdot \left(\sec^2\left(\frac{x}{2}\right) \cdot \frac{1}{2}\right) \]
\[ = \tan\left(\frac{x}{2}\right) + \frac{x}{2} \sec^2\left(\frac{x}{2}\right) \]

### Step 2: Substitute \(\frac{dy}{dx}\) into the expression

Now we substitute \(\frac{dy}{dx}\) into the expression \((1 + \cos x) \frac{dy}{dx} - \sin x\):
\[ (1 + \cos x) \left(\tan\left(\frac{x}{2}\right) + \frac{x}{2} \sec^2\left(\frac{x}{2}\right)\right) - \sin x \]

### Step 3: Simplify the expression

Distributing \((1 + \cos x)\):
\[ (1 + \cos x) \tan\left(\frac{x}{2}\right) + (1 + \cos x) \frac{x}{2} \sec^2\left(\frac{x}{2}\right) - \sin x \]

### Step 4: Use the identity for \(1 + \cos x\)

Recall the identity:
\[ 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \]

So we can rewrite:
\[ 2 \cos^2\left(\frac{x}{2}\right) \tan\left(\frac{x}{2}\right) + 2 \cos^2\left(\frac{x}{2}\right) \frac{x}{2} \sec^2\left(\frac{x}{2}\right) - \sin x \]

### Step 5: Simplify further

Using \(\tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)}\) and \(\sec^2\left(\frac{x}{2}\right) = \frac{1}{\cos^2\left(\frac{x}{2}\right)}\):
\[ = 2 \cos^2\left(\frac{x}{2}\right) \cdot \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} + 2 \cos^2\left(\frac{x}{2}\right) \cdot \frac{x}{2} \cdot \frac{1}{\cos^2\left(\frac{x}{2}\right)} - \sin x \]

This simplifies to:
\[ 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) + x - \sin x \]

Using the double angle identity:
\[ = \sin x + x - \sin x = x \]

### Final Answer
Thus, the value of \((1 + \cos x) \frac{dy}{dx} - \sin x\) is:
\[ \boxed{x} \]