What is the simplified value of `((2)/("cot"(A)/(2)+"tan"(A)/(2)))` ?

Options

A. sin A
B. `"cos"(A)/(2)`
C. `cos^(2)A`
D. `2"sin"(A)/(2)`

Answer

To simplify the expression \(\frac{2}{\frac{\cot(A)}{2} + \frac{\tan(A)}{2}}\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Expression:**
\[ \frac{2}{\frac{\cot(A)}{2} + \frac{\tan(A)}{2}} = \frac{2}{\frac{1}{\tan(A)/2} + \frac{\tan(A)}{2}} \]

2. **Substitute Cotangent:**
Recall that \(\cot(A) = \frac{1}{\tan(A)}\). Therefore, we can rewrite \(\cot(A/2)\) as:
\[ \cot(A/2) = \frac{1}{\tan(A/2)} \]
Thus, we have:
\[ \frac{2}{\frac{1}{\tan(A/2)} + \tan(A/2)} \]

3. **Find a Common Denominator:**
The common denominator for the terms in the denominator is \(\tan(A/2)\):
\[ \frac{1}{\tan(A/2)} + \tan(A/2) = \frac{1 + \tan^2(A/2)}{\tan(A/2)} \]

4. **Use the Pythagorean Identity:**
We know from trigonometric identities that:
\[ 1 + \tan^2(A/2) = \sec^2(A/2) \]
Therefore, we can rewrite the denominator:
\[ \frac{2}{\frac{\sec^2(A/2)}{\tan(A/2)}} \]

5. **Simplify the Expression:**
Now, simplifying further:
\[ = 2 \cdot \frac{\tan(A/2)}{\sec^2(A/2)} = 2 \cdot \tan(A/2) \cdot \cos^2(A/2) \]

6. **Substitute Tan and Cosine:**
Recall that \(\tan(A/2) = \frac{\sin(A/2)}{\cos(A/2)}\):
\[ = 2 \cdot \frac{\sin(A/2)}{\cos(A/2)} \cdot \cos^2(A/2) \]

7. **Final Simplification:**
\[ = 2 \cdot \sin(A/2) \cdot \cos(A/2) \]

8. **Use the Double Angle Identity:**
We know that \(2 \sin(x) \cos(x) = \sin(2x)\):
\[ = \sin(A) \]

### Final Answer:
Thus, the simplified value of \(\frac{2}{\frac{\cot(A)}{2} + \frac{\tan(A)}{2}}\) is:
\[ \sin(A) \]