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 solve 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
We start with the given equation:
\[ \sin \theta \cos \theta \cos 2\theta \cos 4\theta = \frac{1}{2} \]

### Step 2: Use trigonometric identities
We can use the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \) to rewrite the left-hand side:
\[ \frac{1}{2} \sin 2\theta \cos 2\theta \cos 4\theta = \frac{1}{2} \]
Multiplying both sides by 2 gives:
\[ \sin 2\theta \cos 2\theta \cos 4\theta = 1 \]

### Step 3: Further simplify using another identity
Next, we can use the identity \( \sin 2\theta \cos 2\theta = \frac{1}{2} \sin 4\theta \):
\[ \frac{1}{2} \sin 4\theta \cos 4\theta = 1 \]
Multiplying both sides by 2 results in:
\[ \sin 4\theta \cos 4\theta = 2 \]

### Step 4: Use another identity for sine and cosine
Using the identity \( \sin 4\theta \cos 4\theta = \frac{1}{2} \sin 8\theta \):
\[ \frac{1}{2} \sin 8\theta = 2 \]
Multiplying both sides by 2 gives:
\[ \sin 8\theta = 4 \]

### Step 5: Analyze the sine function
The sine function has a maximum value of 1. Therefore, the equation \( \sin 8\theta = 4 \) has no solutions since the sine of any angle cannot exceed 1.

### Conclusion
Since there are no solutions to the equation in the interval \( [0, \pi] \), the number of solutions is:
\[ \text{Number of solutions} = 0 \]