`cot ((A)/(2))-tan((A)/(2))=` A)`2 sin A`B)`2 cos A`C)`2 tan A`D)`2 cot A`

Options

A. `2 sin A`
B. `2 cos A`
C. `2 tan A`
D. `2 cot A`

Answer

To solve the expression \( \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) \), we will follow these steps:

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot\left(\frac{A}{2}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
\[ \tan\left(\frac{A}{2}\right) = \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} \]

### Step 2: Substitute these into the expression
Substituting these definitions into the expression gives:
\[ \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} - \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} \]

### Step 3: Find a common denominator
The common denominator for the two fractions is \( \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \). Thus, we can rewrite the expression as:
\[ \frac{\cos^2\left(\frac{A}{2}\right) - \sin^2\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \]

### Step 4: Use trigonometric identities
Using the identity \( \cos^2\theta - \sin^2\theta = \cos(2\theta) \), we can simplify the numerator:
\[ \cos^2\left(\frac{A}{2}\right) - \sin^2\left(\frac{A}{2}\right) = \cos(A) \]
Thus, the expression becomes:
\[ \frac{\cos(A)}{\sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \]

### Step 5: Simplify the denominator using the double angle identity
Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we have:
\[ \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
So, we can rewrite the expression as:
\[ \frac{\cos(A)}{\frac{1}{2} \sin(A)} = \frac{2 \cos(A)}{\sin(A)} \]

### Step 6: Rewrite in terms of cotangent
Finally, we can express this as:
\[ 2 \cot(A) \]

### Final Answer
Thus, the value of \( \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) \) is:
\[ \boxed{2 \cot(A)} \]