Solve `cot(x//2)-cosec (x//2)=cot x`.

Explanation

To solve the equation \( \cot\left(\frac{x}{2}\right) - \csc\left(\frac{x}{2}\right) = \cot x \), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
We start by rewriting the cotangent and cosecant in terms of sine and cosine:
\[ \cot\left(\frac{x}{2}\right) = \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)}, \quad \csc\left(\frac{x}{2}\right) = \frac{1}{\sin\left(\frac{x}{2}\right)}, \quad \cot x = \frac{\cos x}{\sin x} \]
Thus, we can rewrite the equation as:
\[ \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} - \frac{1}{\sin\left(\frac{x}{2}\right)} = \frac{\cos x}{\sin x} \]

### Step 2: Combine the left side
Now, we can combine the left side over a common denominator:
\[ \frac{\cos\left(\frac{x}{2}\right) - 1}{\sin\left(\frac{x}{2}\right)} = \frac{\cos x}{\sin x} \]

### Step 3: Use the identity for sine
We know that \( \sin x = 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) \). Substituting this into the equation gives:
\[ \frac{\cos\left(\frac{x}{2}\right) - 1}{\sin\left(\frac{x}{2}\right)} = \frac{\cos x}{2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right)} \]

### Step 4: Cancel \(\sin\left(\frac{x}{2}\right)\)
Assuming \( \sin\left(\frac{x}{2}\right) \neq 0 \), we can multiply both sides by \( \sin\left(\frac{x}{2}\right) \):
\[ \cos\left(\frac{x}{2}\right) - 1 = \frac{\cos x}{2 \cos\left(\frac{x}{2}\right)} \]

### Step 5: Cross-multiply
Cross-multiplying gives:
\[ 2 \cos\left(\frac{x}{2}\right) \left(\cos\left(\frac{x}{2}\right) - 1\right) = \cos x \]

### Step 6: Expand and rearrange
Expanding the left side:
\[ 2 \cos^2\left(\frac{x}{2}\right) - 2 \cos\left(\frac{x}{2}\right) = \cos x \]
Using the identity \( \cos x = 2 \cos^2\left(\frac{x}{2}\right) - 1 \), we can substitute:
\[ 2 \cos^2\left(\frac{x}{2}\right) - 2 \cos\left(\frac{x}{2}\right) = 2 \cos^2\left(\frac{x}{2}\right) - 1 \]

### Step 7: Simplify the equation
This simplifies to:
\[ -2 \cos\left(\frac{x}{2}\right) = -1 \]
Thus, we have:
\[ 2 \cos\left(\frac{x}{2}\right) = 1 \implies \cos\left(\frac{x}{2}\right) = \frac{1}{2} \]

### Step 8: Solve for \(x\)
The general solution for \( \cos\left(\frac{x}{2}\right) = \frac{1}{2} \) is:
\[ \frac{x}{2} = 2n\pi \pm \frac{\pi}{3} \]
Multiplying through by 2 gives:
\[ x = 4n\pi \pm \frac{2\pi}{3} \]

### Final Answer
Thus, the solution to the equation is:
\[ x = 4n\pi \pm \frac{2\pi}{3}, \quad n \in \mathbb{Z} \]