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

Options

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

Answer

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

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot \frac{A}{2} = \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} \quad \text{and} \quad \tan \frac{A}{2} = \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} \]
Thus, we can rewrite the expression as:
\[ \frac{2}{\frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} + \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}}} \]

### Step 2: Find a common denominator
To combine the terms in the denominator, we find a common denominator:
\[ \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} + \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} = \frac{\cos^2 \frac{A}{2} + \sin^2 \frac{A}{2}}{\sin \frac{A}{2} \cos \frac{A}{2}} \]

### Step 3: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \cos^2 \frac{A}{2} + \sin^2 \frac{A}{2} = 1 \]
Thus, the denominator simplifies to:
\[ \frac{1}{\sin \frac{A}{2} \cos \frac{A}{2}} \]

### Step 4: Substitute back into the expression
Now substituting back into our expression, we have:
\[ \frac{2}{\frac{1}{\sin \frac{A}{2} \cos \frac{A}{2}}} = 2 \cdot \sin \frac{A}{2} \cos \frac{A}{2} \]

### Step 5: Use the double angle identity
Using the double angle identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ 2 \sin \frac{A}{2} \cos \frac{A}{2} = \sin A \]

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

---