What is the simplified value of `1+cotAcot((A)/(2))`?

Options

A. `cos((A)/(2))`
B. `sin^(2)((A)/(2))`
C. `(1)/(2)"cosec^(2)((A)/(2))`
D. cos A

Answer

To simplify the expression \( 1 + \cot A \cot \left(\frac{A}{2}\right) \), we can follow these steps:

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

### Step 2: Find a common denominator
The common denominator for the expression is \( \sin A \cdot \sin \left(\frac{A}{2}\right) \). So, we rewrite the expression:
\[ 1 + \frac{\cos A \cdot \cos \left(\frac{A}{2}\right)}{\sin A \cdot \sin \left(\frac{A}{2}\right)} = \frac{\sin A \cdot \sin \left(\frac{A}{2}\right) + \cos A \cdot \cos \left(\frac{A}{2}\right)}{\sin A \cdot \sin \left(\frac{A}{2}\right)} \]

### Step 3: Use the cosine addition formula
Using the cosine addition formula:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
We can identify \( B = \frac{A}{2} \) and rewrite the numerator:
\[ \sin A \cdot \sin \left(\frac{A}{2}\right) + \cos A \cdot \cos \left(\frac{A}{2}\right) = \cos \left(A - \frac{A}{2}\right) = \cos \left(\frac{A}{2}\right) \]

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

### Step 5: Simplify further
Using the identity \( \sin A = 2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right) \), we can rewrite the denominator:
\[ \sin A \cdot \sin \left(\frac{A}{2}\right) = 2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right) \cdot \sin \left(\frac{A}{2}\right) = 2 \sin^2 \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right) \]

### Step 6: Final expression
Thus, the expression simplifies to:
\[ \frac{\cos \left(\frac{A}{2}\right)}{2 \sin^2 \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right)} = \frac{1}{2 \sin^2 \left(\frac{A}{2}\right)} \]

### Final Answer
The simplified value of \( 1 + \cot A \cot \left(\frac{A}{2}\right) \) is:
\[ \frac{1}{2 \sin^2 \left(\frac{A}{2}\right)} \]

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