The number of solution of the equation `sin theta cos theta cos 2 theta .cos 4theta =1/2` in the interval `[0,pi]`

Options

A. 0
B. 8
C. 9
D. 10

Answer

To find the number of solutions for the equation \( \sin \theta \cos \theta \cos 2\theta \cos 4\theta = \frac{1}{2} \) in the interval \( [0, \pi] \), we can follow these steps:

### Step 1: Rewrite the Equation
Start with the given equation:
\[ \sin \theta \cos \theta \cos 2\theta \cos 4\theta = \frac{1}{2} \]

### Step 2: Multiply Both Sides by 2
Multiply both sides of the equation by 2:
\[ 2 \sin \theta \cos \theta \cos 2\theta \cos 4\theta = 1 \]

### Step 3: Use the Identity for \( \sin 2\theta \)
Recall the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substitute this into the equation:
\[ \sin 2\theta \cos 2\theta \cos 4\theta = 1 \]

### Step 4: Multiply Both Sides by 2 Again
Multiply both sides by 2:
\[ 2 \sin 2\theta \cos 2\theta \cos 4\theta = 2 \]

### Step 5: Use the Identity for \( \sin 4\theta \)
Using the identity \( \sin 4\theta = 2 \sin 2\theta \cos 2\theta \), we can rewrite the equation:
\[ \sin 4\theta \cos 4\theta = 2 \]

### Step 6: Multiply Both Sides by 2 Once More
Multiply both sides by 2:
\[ 2 \sin 4\theta \cos 4\theta = 4 \]

### Step 7: Use the Identity for \( \sin 8\theta \)
Using the identity \( \sin 8\theta = 2 \sin 4\theta \cos 4\theta \), we can rewrite the equation:
\[ \sin 8\theta = 4 \]

### Step 8: Analyze the Range of the Sine Function
The sine function has a range of \([-1, 1]\). Therefore, the equation \( \sin 8\theta = 4 \) has no solutions because \( 4 \) is outside the range of the sine function.

### Conclusion
Since there are no solutions to the equation \( \sin 8\theta = 4 \), the number of solutions to the original equation \( \sin \theta \cos \theta \cos 2\theta \cos 4\theta = \frac{1}{2} \) in the interval \( [0, \pi] \) is:
\[ \text{Number of solutions} = 0 \]