If `m` and `n` respectively are the numbers of positive and negative values of `theta` in the interval `[-pi,pi]` that satisfy the equation `cos2 theta cos((theta)/(2))=cos3 theta cos((9 theta)/(2)),` then `mn` is equal to

Answer

To solve the equation \( \cos(2\theta) \cos\left(\frac{\theta}{2}\right) = \cos(3\theta) \cos\left(\frac{9\theta}{2}\right) \) and find the values of \( m \) and \( n \), we will follow these steps:

### Step 1: Apply the Product-to-Sum Identities
Using the identity \( 2 \cos A \cos B = \cos(A+B) + \cos(A-B) \), we can rewrite both sides of the equation.

\[ \cos(2\theta) \cos\left(\frac{\theta}{2}\right) = \frac{1}{2} \left( \cos\left(2\theta + \frac{\theta}{2}\right) + \cos\left(2\theta - \frac{\theta}{2}\right) \right) \]

\[ \cos(3\theta) \cos\left(\frac{9\theta}{2}\right) = \frac{1}{2} \left( \cos\left(3\theta + \frac{9\theta}{2}\right) + \cos\left(3\theta - \frac{9\theta}{2}\right) \right) \]

### Step 2: Simplify the Expressions
Now we simplify the arguments of the cosine functions:

For the left side:
\[ \cos\left(2\theta + \frac{\theta}{2}\right) = \cos\left(\frac{5\theta}{2}\right) \]
\[ \cos\left(2\theta - \frac{\theta}{2}\right) = \cos\left(\frac{3\theta}{2}\right) \]

For the right side:
\[ \cos\left(3\theta + \frac{9\theta}{2}\right) = \cos\left(\frac{15\theta}{2}\right) \]
\[ \cos\left(3\theta - \frac{9\theta}{2}\right) = \cos\left(-\frac{3\theta}{2}\right) = \cos\left(\frac{3\theta}{2}\right) \]

### Step 3: Set the Equations Equal
Now we have:
\[ \frac{1}{2} \left( \cos\left(\frac{5\theta}{2}\right) + \cos\left(\frac{3\theta}{2}\right) \right) = \frac{1}{2} \left( \cos\left(\frac{15\theta}{2}\right) + \cos\left(\frac{3\theta}{2}\right) \right) \]

### Step 4: Cancel Common Terms
By canceling \( \frac{1}{2} \) and \( \cos\left(\frac{3\theta}{2}\right) \) from both sides, we get:
\[ \cos\left(\frac{5\theta}{2}\right) = \cos\left(\frac{15\theta}{2}\right) \]

### Step 5: Solve the Cosine Equation
The general solution for \( \cos A = \cos B \) is:
\[ A = 2n\pi \pm B \]
Thus, we have two cases:
1. \( \frac{5\theta}{2} = 2n\pi + \frac{15\theta}{2} \)
2. \( \frac{5\theta}{2} = 2n\pi - \frac{15\theta}{2} \)

### Step 6: Solve for \( \theta \)
From the first case:
\[ \frac{5\theta}{2} - \frac{15\theta}{2} = 2n\pi \implies -5\theta = 2n\pi \implies \theta = -\frac{2n\pi}{5} \]

From the second case:
\[ \frac{5\theta}{2} + \frac{15\theta}{2} = 2n\pi \implies 10\theta = 2n\pi \implies \theta = \frac{n\pi}{5} \]

### Step 7: Count Positive and Negative Solutions
Now we need to find the number of positive and negative values of \( \theta \) in the interval \( [-\pi, \pi] \).

1. For \( \theta = -\frac{2n\pi}{5} \):
- Negative values: \( n = 1, 2, 3, 4, 5 \) gives \( -\frac{2\pi}{5}, -\frac{4\pi}{5}, -\frac{6\pi}{5}, -\frac{8\pi}{5}, -\frac{10\pi}{5} \) (only \( n = 1, 2, 3, 4 \) are valid).
- Total negative solutions \( n = 4 \).

2. For \( \theta = \frac{n\pi}{5} \):
- Positive values: \( n = 1, 2, 3, 4, 5 \) gives \( \frac{2\pi}{5}, \frac{4\pi}{5}, \frac{6\pi}{5}, \frac{8\pi}{5}, \frac{10\pi}{5} \) (only \( n = 1, 2 \) are valid).
- Total positive solutions \( m = 5 \).

### Step 8: Calculate \( mn \)
Finally, we calculate \( mn \):
\[ mn = 5 \times 4 = 20 \]

### Final Answer
The value of \( mn \) is \( 20 \).