If `cos (theta - alpha),cos(theta),Cos(theta + alpha)` are in H.P. and ` cos alpha ne 1`, then angle `alpha` lie in the

Options

A. II quadrant
B. III quadrant
C. I quadrant
D. IV quadrant

Answer

To solve the problem, we need to determine the quadrant in which the angle α lies, given that \( \cos(\theta - \alpha), \cos(\theta), \cos(\theta + \alpha) \) are in Harmonic Progression (H.P.) and \( \cos \alpha \neq 1 \).

### Step-by-Step Solution:

1. **Understanding H.P.**:
If three numbers \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.). Therefore, we can write:
\[ \frac{1}{\cos(\theta - \alpha)}, \frac{1}{\cos(\theta)}, \frac{1}{\cos(\theta + \alpha)} \]
are in A.P.

2. **Setting Up the A.P. Condition**:
For three terms to be in A.P., the condition is:
\[ 2 \cdot \frac{1}{\cos(\theta)} = \frac{1}{\cos(\theta - \alpha)} + \frac{1}{\cos(\theta + \alpha)} \]

3. **Cross-Multiplying**:
This leads to:
\[ 2 \cos(\theta - \alpha) \cos(\theta + \alpha) = \cos(\theta) \left( \cos(\theta + \alpha) + \cos(\theta - \alpha) \right) \]

4. **Using the Cosine Addition Formula**:
The right-hand side can be simplified using the cosine addition formula:
\[ \cos(\theta + \alpha) + \cos(\theta - \alpha) = 2 \cos(\theta) \cos(\alpha) \]
Thus, we have:
\[ 2 \cos(\theta - \alpha) \cos(\theta + \alpha) = 2 \cos^2(\theta) \cos(\alpha) \]

5. **Simplifying the Equation**:
Dividing both sides by 2 (assuming \( \cos(\theta) \neq 0 \)):
\[ \cos(\theta - \alpha) \cos(\theta + \alpha) = \cos^2(\theta) \cos(\alpha) \]

6. **Using the Product-to-Sum Formula**:
The left-hand side can be expressed as:
\[ \frac{1}{2} \left( \cos(2\theta) + \cos(2\alpha) \right) \]
Thus, we have:
\[ \frac{1}{2} \left( \cos(2\theta) + \cos(2\alpha) \right) = \cos^2(\theta) \cos(\alpha) \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ \cos(2\theta) + \cos(2\alpha) = 2 \cos^2(\theta) \cos(\alpha) \]

8. **Analyzing the Range of \( \cos(\alpha) \)**:
We know that \( \cos(\alpha) \) must be in the range that satisfies \( \cos(\alpha) \neq 1 \). Since \( \cos(\alpha) \) is negative, we can conclude that:
\[ -1 < \cos(\alpha) < 0 \]

9. **Determining the Quadrant**:
The cosine function is negative in the second and third quadrants. Therefore, angle \( \alpha \) must lie in either the second quadrant (90° to 180°) or the third quadrant (180° to 270°).

### Conclusion:
The angle \( \alpha \) lies in the **second and third quadrants**.