Slove : `sin9theta =sintheta `

Answer

To solve the equation \( \sin 9\theta = \sin \theta \), we can use the property of sine functions. The general solution for \( \sin x = \sin y \) is given by:

\[ x = n\pi + (-1)^n y \]

where \( n \) is any integer. In our case, we can set \( x = 9\theta \) and \( y = \theta \).

### Step 1: Set up the equation
Using the general solution, we have:

\[ 9\theta = n\pi + (-1)^n \theta \]

### Step 2: Rearrange the equation
We can rearrange this equation into two cases based on the parity of \( n \) (whether \( n \) is even or odd).

### Case 1: \( n = 2m \) (even)
Substituting \( n = 2m \):

\[ 9\theta = 2m\pi + \theta \]

Subtracting \( \theta \) from both sides gives:

\[ 9\theta - \theta = 2m\pi \]

This simplifies to:

\[ 8\theta = 2m\pi \]

Dividing both sides by 8:

\[ \theta = \frac{m\pi}{4} \]

### Case 2: \( n = 2m + 1 \) (odd)
Substituting \( n = 2m + 1 \):

\[ 9\theta = (2m + 1)\pi - \theta \]

Adding \( \theta \) to both sides gives:

\[ 9\theta + \theta = (2m + 1)\pi \]

This simplifies to:

\[ 10\theta = (2m + 1)\pi \]

Dividing both sides by 10:

\[ \theta = \frac{(2m + 1)\pi}{10} \]

### Final Solution
Thus, the solutions for \( \theta \) are:

1. From Case 1: \( \theta = \frac{m\pi}{4} \) for \( m \in \mathbb{Z} \)
2. From Case 2: \( \theta = \frac{(2m + 1)\pi}{10} \) for \( m \in \mathbb{Z} \)