Let `S_(1)` be the set of all those solution of the equation `(1+a) cos theta cos(2 theta-b)=(1+a cos 2 theta) cos (theta -b)`
which are independent of a and b and `S_(2)` be the set of all such solutions which are dependent on a and b. Then
The set `S_(1)` and `S_(2)` are

Options

A. `{n pi, n in Z}` and `1/2 {n pi +(-1)^(n) sin^(-1) (a sin b) +b, n in Z}`
B. `{n pi/2, n in Z}` and `{n pi+(-1)^(n) sin^(-1) (a sin b), n in Z}`
C. `{n pi/2, n in Z}` and `{n pi +(-1)^(n) sin^(-1) (a/2 sin b), n in Z}`
D. none of these

Explanation

To solve the equation \((1 + a) \cos \theta \cos(2\theta - b) = (1 + a \cos 2\theta) \cos(\theta - b)\), we will follow these steps:

### Step 1: Expand both sides of the equation
We start by expanding the left-hand side and the right-hand side of the equation.

**Left-hand side:**
\[ (1 + a) \cos \theta \cos(2\theta - b) = (1 + a) \left( \frac{1}{2} (\cos(\theta + (2\theta - b)) + \cos(\theta - (2\theta - b))) \right) \]
Using the cosine addition formula, we can express this as:
\[ = \frac{(1 + a)}{2} \left( \cos(3\theta - b) + \cos(\theta + b) \right) \]

**Right-hand side:**
\[ (1 + a \cos 2\theta) \cos(\theta - b) = \cos(\theta - b) + a \cos 2\theta \cos(\theta - b) \]
Using the cosine addition formula again:
\[ = \cos(\theta - b) + a \left( \frac{1}{2} (\cos(3\theta - b) + \cos(b - \theta)) \right) \]

### Step 2: Set the two sides equal
Now we set the two expanded sides equal to each other:
\[ \frac{(1 + a)}{2} \left( \cos(3\theta - b) + \cos(\theta + b) \right) = \cos(\theta - b) + a \left( \frac{1}{2} (\cos(3\theta - b) + \cos(b - \theta)) \right) \]

### Step 3: Simplify the equation
Rearranging the equation gives us:
\[ \frac{(1 + a)}{2} \cos(3\theta - b) + \frac{(1 + a)}{2} \cos(\theta + b) - \cos(\theta - b) - \frac{a}{2} \cos(3\theta - b) - \frac{a}{2} \cos(b - \theta) = 0 \]
This can be grouped and simplified further.

### Step 4: Identify solutions independent of \(a\) and \(b\)
To find the solutions independent of \(a\) and \(b\), we look for values of \(\theta\) that satisfy the equation regardless of the parameters.

1. **Condition 1:** \( \sin \theta = 0 \) leads to:
\[ \theta = n\pi, \quad n \in \mathbb{Z} \]

2. **Condition 2:** \( 2\theta - b = n\pi \) leads to:
\[ \theta = \frac{n\pi + b}{2}, \quad n \in \mathbb{Z} \]

### Step 5: Define sets \(S_1\) and \(S_2\)
- **Set \(S_1\)**: Solutions independent of \(a\) and \(b\):
\[ S_1 = \{ n\pi \mid n \in \mathbb{Z} \} \]

- **Set \(S_2\)**: Solutions dependent on \(a\) and \(b\):
\[ S_2 = \left\{ \frac{n\pi + b}{2} \mid n \in \mathbb{Z} \right\} \]

### Conclusion
The sets \(S_1\) and \(S_2\) are defined as above, where \(S_1\) contains solutions that do not depend on parameters \(a\) and \(b\), while \(S_2\) contains solutions that do depend on these parameters.