In a `Delta ABC`, the minimum value of
`sec^(2). (A)/(2)+sec^(2). (B)/(2)+sec^(2). (C)/(2)` is equal to

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the minimum value of the expression \(\sec^2\left(\frac{A}{2}\right) + \sec^2\left(\frac{B}{2}\right) + \sec^2\left(\frac{C}{2}\right)\) in a triangle \(ABC\), we can follow these steps:

### Step 1: Understand the angles in a triangle
In any triangle, the sum of the internal angles is given by:
\[ A + B + C = 180^\circ \]

### Step 2: Express the angles in terms of a single variable
Since we are looking for the minimum value of the expression, we can assume that the angles are equal for symmetry. Therefore, let:
\[ A = B = C = 60^\circ \]
This is valid because \(60^\circ + 60^\circ + 60^\circ = 180^\circ\).

### Step 3: Substitute the angles into the expression
Now, substituting \(A\), \(B\), and \(C\) into the expression:
\[ \sec^2\left(\frac{A}{2}\right) + \sec^2\left(\frac{B}{2}\right) + \sec^2\left(\frac{C}{2}\right) = \sec^2\left(\frac{60^\circ}{2}\right) + \sec^2\left(\frac{60^\circ}{2}\right) + \sec^2\left(\frac{60^\circ}{2}\right) \]
This simplifies to:
\[ 3 \cdot \sec^2(30^\circ) \]

### Step 4: Calculate \(\sec^2(30^\circ)\)
We know that:
\[ \sec(30^\circ) = \frac{1}{\cos(30^\circ)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
Thus:
\[ \sec^2(30^\circ) = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]

### Step 5: Calculate the total value
Now, substituting back into the expression:
\[ 3 \cdot \sec^2(30^\circ) = 3 \cdot \frac{4}{3} = 4 \]

### Conclusion
Therefore, the minimum value of \(\sec^2\left(\frac{A}{2}\right) + \sec^2\left(\frac{B}{2}\right) + \sec^2\left(\frac{C}{2}\right)\) is:
\[ \boxed{4} \]