Solve the equation `cot((theta )/(2))-"cosec"((theta)/2)=cot theta `

Answer

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

### Step 1: Rewrite the equation using trigonometric identities
We start with the equation:
\[ \cot\left(\frac{\theta}{2}\right) - \csc\left(\frac{\theta}{2}\right) = \cot(\theta) \]
Recall that:
\[ \cot\left(\frac{\theta}{2}\right) = \frac{\cos\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \quad \text{and} \quad \csc\left(\frac{\theta}{2}\right) = \frac{1}{\sin\left(\frac{\theta}{2}\right)} \]
Thus, we can rewrite the left side:
\[ \frac{\cos\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} - \frac{1}{\sin\left(\frac{\theta}{2}\right)} = \frac{\cos\left(\frac{\theta}{2}\right) - 1}{\sin\left(\frac{\theta}{2}\right)} \]
So the equation becomes:
\[ \frac{\cos\left(\frac{\theta}{2}\right) - 1}{\sin\left(\frac{\theta}{2}\right)} = \cot(\theta) \]

### Step 2: Rewrite \( \cot(\theta) \)
Using the identities for \( \cot(\theta) \):
\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]
And we know that:
\[ \cos(\theta) = 2\cos^2\left(\frac{\theta}{2}\right) - 1 \quad \text{and} \quad \sin(\theta) = 2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) \]
Thus, we can write:
\[ \cot(\theta) = \frac{2\cos^2\left(\frac{\theta}{2}\right) - 1}{2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)} \]

### Step 3: Set the two sides equal
Now we have:
\[ \frac{\cos\left(\frac{\theta}{2}\right) - 1}{\sin\left(\frac{\theta}{2}\right)} = \frac{2\cos^2\left(\frac{\theta}{2}\right) - 1}{2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)} \]
Cross-multiplying gives:
\[ (\cos\left(\frac{\theta}{2}\right) - 1) \cdot 2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) = (2\cos^2\left(\frac{\theta}{2}\right) - 1) \cdot \sin\left(\frac{\theta}{2}\right) \]

### Step 4: Simplify the equation
Cancel \( \sin\left(\frac{\theta}{2}\right) \) from both sides (assuming \( \sin\left(\frac{\theta}{2}\right) \neq 0 \)):
\[ 2\cos\left(\frac{\theta}{2}\right)(\cos\left(\frac{\theta}{2}\right) - 1) = 2\cos^2\left(\frac{\theta}{2}\right) - 1 \]
Expanding the left side:
\[ 2\cos^2\left(\frac{\theta}{2}\right) - 2\cos\left(\frac{\theta}{2}\right) = 2\cos^2\left(\frac{\theta}{2}\right) - 1 \]
Now, simplifying gives:
\[ -2\cos\left(\frac{\theta}{2}\right) = -1 \]
Thus:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{1}{2} \]

### Step 5: Solve for \( \theta \)
The general solution for \( \cos\left(\frac{\theta}{2}\right) = \frac{1}{2} \) is:
\[ \frac{\theta}{2} = 2n\pi \pm \frac{\pi}{3} \]
Multiplying by 2 gives:
\[ \theta = 4n\pi \pm \frac{2\pi}{3} \]

### Final Solution
Thus, the solutions to the equation are:
\[ \theta = 4n\pi + \frac{2\pi}{3} \quad \text{and} \quad \theta = 4n\pi - \frac{2\pi}{3} \]
where \( n \) is any integer.

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