Find the number of solution for , `sin 5 theta * cos 3 theta = sin 9 theta * cos 7 theta ` in `[0,(pi)/2]`

Answer

To solve the equation \( \sin(5\theta) \cos(3\theta) = \sin(9\theta) \cos(7\theta) \) in the interval \( [0, \frac{\pi}{2}] \), we can follow these steps:

### Step 1: Rewrite the Equation
The given equation can be rewritten using the product-to-sum identities:
\[ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \]
Applying this to both sides:
\[ \sin(5\theta) \cos(3\theta) = \frac{1}{2} [\sin(8\theta) + \sin(2\theta)] \]
\[ \sin(9\theta) \cos(7\theta) = \frac{1}{2} [\sin(16\theta) + \sin(2\theta)] \]
Thus, the equation becomes:
\[ \frac{1}{2} [\sin(8\theta) + \sin(2\theta)] = \frac{1}{2} [\sin(16\theta) + \sin(2\theta)] \]

### Step 2: Simplify the Equation
We can eliminate the \( \frac{1}{2} \) and \( \sin(2\theta) \) from both sides:
\[ \sin(8\theta) = \sin(16\theta) \]

### Step 3: Solve the Sine Equation
The general solution for \( \sin A = \sin B \) is given by:
\[ A = B + 2n\pi \quad \text{or} \quad A = \pi - B + 2n\pi \]
Applying this to our equation:
1. \( 8\theta = 16\theta + 2n\pi \)
2. \( 8\theta = \pi - 16\theta + 2n\pi \)

#### From the first equation:
\[ 8\theta - 16\theta = 2n\pi \implies -8\theta = 2n\pi \implies \theta = -\frac{n\pi}{4} \]
This does not yield any solutions in the interval \( [0, \frac{\pi}{2}] \).

#### From the second equation:
\[ 8\theta + 16\theta = \pi + 2n\pi \implies 24\theta = \pi + 2n\pi \implies \theta = \frac{(1 + 2n)\pi}{24} \]

### Step 4: Find Valid Solutions
Now we need to find valid values of \( n \) such that \( \theta \) lies in the interval \( [0, \frac{\pi}{2}] \):
\[ 0 \leq \frac{(1 + 2n)\pi}{24} \leq \frac{\pi}{2} \]
This simplifies to:
\[ 0 \leq 1 + 2n \leq 12 \implies -1 \leq 2n \leq 11 \implies 0 \leq n \leq 5.5 \]
Thus, \( n \) can take the values \( 0, 1, 2, 3, 4, 5 \).

### Step 5: Calculate the Solutions
Calculating \( \theta \) for each valid \( n \):
- For \( n = 0 \): \( \theta = \frac{\pi}{24} \)
- For \( n = 1 \): \( \theta = \frac{3\pi}{24} = \frac{\pi}{8} \)
- For \( n = 2 \): \( \theta = \frac{5\pi}{24} \)
- For \( n = 3 \): \( \theta = \frac{7\pi}{24} \)
- For \( n = 4 \): \( \theta = \frac{9\pi}{24} = \frac{3\pi}{8} \)
- For \( n = 5 \): \( \theta = \frac{11\pi}{24} \)

### Step 6: Count the Solutions
The solutions in the interval \( [0, \frac{\pi}{2}] \) are:
1. \( \frac{\pi}{24} \)
2. \( \frac{\pi}{8} \)
3. \( \frac{5\pi}{24} \)
4. \( \frac{7\pi}{24} \)
5. \( \frac{3\pi}{8} \)
6. \( \frac{11\pi}{24} \)

Thus, the total number of solutions is **6**.

### Final Answer
The total number of solutions for the equation \( \sin(5\theta) \cos(3\theta) = \sin(9\theta) \cos(7\theta) \) in the interval \( [0, \frac{\pi}{2}] \) is **6**.