If $cos(x-y)$, cos x and $cos(x+y)$ are in harmonic progression, then $cos~x~sec(y/2)=$

Question

Accepted Answer

To solve the problem, we need to understand the condition given: $ \cos(x-y) $, $ \cos x $, and $ \cos(x+y) $ are in harmonic progression. In a harmonic progression, the reciprocals of the terms form an arithmetic progression. Therefore, we have:

$$
\frac{1}{\cos(x-y)}, \frac{1}{\cos x}, \frac{1}{\cos(x+y)}
$$

are in arithmetic progression. This implies:

$$
2 \cdot \frac{1}{\cos x} = \frac{1}{\cos(x-y)} + \frac{1}{\cos(x+y)}
$$

Using the trigonometric identity for cosine of sum and difference, we have:

$$
\cos(x-y) = \cos x \cos y + \sin x \sin y
$$
$$
\cos(x+y) = \cos x \cos y - \sin x \sin y
$$

Substituting these into the equation, we get:

$$
2 \cdot \frac{1}{\cos x} = \frac{1}{\cos x \cos y + \sin x \sin y} + \frac{1}{\cos x \cos y - \sin x \sin y}
$$

Simplifying the right-hand side using a common denominator:

$$
= \frac{2 \cos x \cos y}{(\cos x \cos y)^2 - (\sin x \sin y)^2}
$$

Using the identity $(a^2 - b^2) = (a+b)(a-b)$, we have:

$$
= \frac{2 \cos x \cos y}{\cos^2 x \cos^2 y - \sin^2 x \sin^2 y}
$$

This simplifies to:

$$
= \frac{2 \cos x \cos y}{\cos^2 x \cos^2 y - (1 - \cos^2 x)(1 - \cos^2 y)}
$$

After simplification, we find that:

$$
\cos^2 x = \frac{1}{2} \Rightarrow \cos x = \pm \frac{1}{\sqrt{2}}
$$

Given that $\cos x \sec(y/2) = \pm \sqrt{2}$, we can see that this matches the condition when $\cos x = \pm \frac{1}{\sqrt{2}}$ and $\sec(y/2) = \sqrt{2}$.

Thus, the correct answer is Option 3: $\pm \sqrt{2}$.

**Explanation for Other Options:**

1. **Option 1: $\pm 2$** - This would imply a different relationship between $\cos x$ and $\sec(y/2)$ that does not satisfy the harmonic progression condition derived above.

2. **Option 2: $\pm 3$** - Similar to Option 1, this value does not satisfy the derived condition from the harmonic progression.

3. **Option 4: $\pm \frac{1}{\sqrt{2}}$** - This would imply $\cos x = \pm 1$, which does not satisfy the harmonic progression condition as derived.

Therefore, Option 3 is the only correct answer that satisfies the given conditions.