Solutions of the equations `"cos"^(2) ((1)/(2) px)+ "cos"^(2) ((1)/(2) qx) = 1` form an arithmetic progression with common difference

Options

A. `(2)/(p+q)`
B. `(2)/(p-q)`
C. `(pi)/(p+q)`
D. none of these

Answer

To solve the equation \( \cos^2\left(\frac{p}{2} x\right) + \cos^2\left(\frac{q}{2} x\right) = 1 \) and find the common difference of the solutions that form an arithmetic progression (AP), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos^2\left(\frac{p}{2} x\right) + \cos^2\left(\frac{q}{2} x\right) = 1 \]
Using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we can rewrite the equation as:
\[ \cos^2\left(\frac{p}{2} x\right) = 1 - \cos^2\left(\frac{q}{2} x\right) = \sin^2\left(\frac{q}{2} x\right) \]

### Step 2: Set up the equation
This leads to:
\[ \cos^2\left(\frac{p}{2} x\right) = \sin^2\left(\frac{q}{2} x\right) \]
Taking the square root of both sides, we have:
\[ \cos\left(\frac{p}{2} x\right) = \pm \sin\left(\frac{q}{2} x\right) \]

### Step 3: Use trigonometric identities
Using the identity \( \sin\theta = \cos\left(\frac{\pi}{2} - \theta\right) \), we can express the equation as:
\[ \cos\left(\frac{p}{2} x\right) = \cos\left(\frac{\pi}{2} - \frac{q}{2} x\right) \]
This gives us two cases to consider:
1. \( \frac{p}{2} x = n\pi + \left(\frac{\pi}{2} - \frac{q}{2} x\right) \)
2. \( \frac{p}{2} x = n\pi - \left(\frac{\pi}{2} - \frac{q}{2} x\right) \)

### Step 4: Solve for \( x \)
For the first case:
\[ \frac{p}{2} x + \frac{q}{2} x = n\pi + \frac{\pi}{2} \]
This simplifies to:
\[ \frac{(p + q)}{2} x = n\pi + \frac{\pi}{2} \]
Thus,
\[ x = \frac{2(n\pi + \frac{\pi}{2})}{p + q} = \frac{2n\pi + \pi}{p + q} = \frac{(2n + 1)\pi}{p + q} \]

For the second case:
\[ \frac{p}{2} x - \frac{q}{2} x = n\pi - \frac{\pi}{2} \]
This simplifies to:
\[ \frac{(p - q)}{2} x = n\pi - \frac{\pi}{2} \]
Thus,
\[ x = \frac{2(n\pi - \frac{\pi}{2})}{p - q} = \frac{2n\pi - \pi}{p - q} = \frac{(2n - 1)\pi}{p - q} \]

### Step 5: Identify the common difference
The solutions of the first case are:
\[ x_n = \frac{(2n + 1)\pi}{p + q} \]
The common difference \( D \) can be calculated as:
\[ D = x_{n+1} - x_n = \frac{(2(n+1) + 1)\pi}{p + q} - \frac{(2n + 1)\pi}{p + q} = \frac{(2n + 3)\pi - (2n + 1)\pi}{p + q} = \frac{2\pi}{p + q} \]

### Conclusion
Thus, the common difference of the arithmetic progression formed by the solutions of the given equation is:
\[ \boxed{\frac{2\pi}{p + q}} \]