If `a^(2),b^(2),c^(2)` are in A.P.,then which of the following is also in A.P.?

Options

A. sin A , sin B, sin C
B. tan A , tan B, tan C
C. cot A , cot B , cot C
D. none of these

Answer

To solve the problem, we need to determine which of the following expressions is also in Arithmetic Progression (A.P.) if \(a^2\), \(b^2\), and \(c^2\) are in A.P.

### Step-by-Step Solution:

1. **Understanding A.P.**:
By definition, three numbers \(x\), \(y\), and \(z\) are in A.P. if \(2y = x + z\). For our case, since \(a^2\), \(b^2\), and \(c^2\) are in A.P., we can write:
\[ 2b^2 = a^2 + c^2 \]

2. **Using the Sine Rule**:
In a triangle, the sides are proportional to the sines of their opposite angles. Thus, we can express:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \quad \text{(some constant)} \]
From this, we can derive:
\[ \sin A = \frac{a}{k}, \quad \sin B = \frac{b}{k}, \quad \sin C = \frac{c}{k} \]

3. **Relating Sine Values**:
If \(a^2\), \(b^2\), and \(c^2\) are in A.P., then it can be shown that \(\sin A\), \(\sin B\), and \(\sin C\) are also in A.P. This is because the ratios of the sides to the sine of their angles maintain the same proportional relationship.

4. **Considering Cotangent**:
The cotangent function is the reciprocal of the tangent function. If \( \cot A\), \( \cot B\), and \( \cot C\) are in A.P., then it implies that:
\[ 2 \cot B = \cot A + \cot C \]
This can be derived from the properties of cotangent and the relationship established from the angles.

5. **Conclusion**:
Since \(a^2\), \(b^2\), and \(c^2\) are in A.P., it follows that \( \cot A\), \( \cot B\), and \( \cot C\) are also in A.P. Therefore, we conclude that if \(a^2\), \(b^2\), and \(c^2\) are in A.P., then \( \cot A\), \( \cot B\), and \( \cot C\) are also in A.P.

### Final Answer:
Thus, the answer is that \( \cot A\), \( \cot B\), and \( \cot C\) are in A.P.