If the perimeter of a triangle ABC is 30cm, then what is the value of a `cos^(2)(C//2)+c cos^(2)(A//2)`?

Options

A. 15cm
B. 10 cm
C. `(15)/(2)` cm
D. 13 cm

Answer

To solve the problem, we need to find the value of \( A \cos^2\left(\frac{C}{2}\right) + C \cos^2\left(\frac{A}{2}\right) \) given that the perimeter of triangle ABC is 30 cm.

### Step-by-Step Solution:

1. **Understanding the Perimeter**:
The perimeter of triangle ABC is given by:
\[ A + B + C = 30 \text{ cm} \]
where \( A \), \( B \), and \( C \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \) respectively.

2. **Finding the Semi-Perimeter**:
The semi-perimeter \( s \) of the triangle is given by:
\[ s = \frac{A + B + C}{2} = \frac{30}{2} = 15 \text{ cm} \]

3. **Using the Cosine Half-Angle Formulas**:
We know the formulas for \( \cos\left(\frac{A}{2}\right) \) and \( \cos\left(\frac{C}{2}\right) \):
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{s(s - A)}{BC}} \]
\[ \cos\left(\frac{C}{2}\right) = \sqrt{\frac{s(s - C)}{AB}} \]

4. **Calculating \( \cos^2\left(\frac{A}{2}\right) \) and \( \cos^2\left(\frac{C}{2}\right) \)**:
We can express \( \cos^2\left(\frac{A}{2}\right) \) and \( \cos^2\left(\frac{C}{2}\right) \) as:
\[ \cos^2\left(\frac{A}{2}\right) = \frac{s(s - A)}{BC} \]
\[ \cos^2\left(\frac{C}{2}\right) = \frac{s(s - C)}{AB} \]

5. **Substituting into the Expression**:
Now we substitute these into the expression we need to evaluate:
\[ A \cos^2\left(\frac{C}{2}\right) + C \cos^2\left(\frac{A}{2}\right) = A \cdot \frac{s(s - C)}{AB} + C \cdot \frac{s(s - A)}{BC} \]

6. **Simplifying the Expression**:
Since \( s = 15 \), we can substitute \( s \) into the equation:
\[ = A \cdot \frac{15(15 - C)}{AB} + C \cdot \frac{15(15 - A)}{BC} \]
However, we can also notice that \( A + B + C = 30 \) implies that \( B = 30 - A - C \).

7. **Final Calculation**:
After substituting and simplifying, we find that the expression simplifies to a constant value. In this case, it can be shown that:
\[ A \cos^2\left(\frac{C}{2}\right) + C \cos^2\left(\frac{A}{2}\right) = 15 \]

### Conclusion:
Thus, the value of \( A \cos^2\left(\frac{C}{2}\right) + C \cos^2\left(\frac{A}{2}\right) \) is \( 15 \).