The product of common differences of all possible AP which are made from values of 'x' satisfying `cos^(2)((1)/(2)lambda x )+ cos^(2)((1)/(2) mu x )=1 `

Options

A. `(4 pi^(2))/(lambda^(2)-mu^(2))`
B. `(4pi)/( lambda - mu ) `
C. `(2 pi^(2))/(lambda^(2)-mu^(2))`
D. None of these

Answer

To solve the problem, we need to find the product of the common differences of all possible arithmetic progressions (APs) formed from the values of \( x \) that satisfy the equation:

\[ \cos^2\left(\frac{1}{2} \lambda x\right) + \cos^2\left(\frac{1}{2} \mu x\right) = 1 \]

### Step 1: Transform the equation

We start by rewriting the equation:

\[ \cos^2\left(\frac{1}{2} \lambda x\right) = 1 - \cos^2\left(\frac{1}{2} \mu x\right) = \sin^2\left(\frac{1}{2} \mu x\right) \]