Let `f_(n)(theta)= 2 sin . (theta)/(2) sin. (3theta)/(2) + 2 sin.(theta)/(2) sin. (5theta)/(2) + 2sin. (theta)/(2) sin. (7 theta)/(2) + ... + 2 sin (2 n+1) (theta)/(2), n in N`,
then which of the following is/are correct ?

Options

A. `f_(9)(pi/4)=(1)/(sqrt(2))`
B. `f_(n)((2pi)/(n))=0, n in N`
C. `f_(3)((2pi)/(7))=0`
D. `f_(9)(pi/4)=-(1)/(sqrt(2))`

Answer

To solve the problem, we need to analyze the function \( f_n(\theta) \) given by:

\[ f_n(\theta) = 2 \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right) + 2 \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{5\theta}{2}\right) + 2 \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{7\theta}{2}\right) + \ldots + 2 \sin\left(\frac{(2n+1)\theta}{2}\right) \]

### Step 1: Factor out common terms
We can factor out \( 2 \sin\left(\frac{\theta}{2}\right) \) from the entire expression:

\[ f_n(\theta) = 2 \sin\left(\frac{\theta}{2}\right) \left( \sin\left(\frac{3\theta}{2}\right) + \sin\left(\frac{5\theta}{2}\right) + \sin\left(\frac{7\theta}{2}\right) + \ldots + \sin\left(\frac{(2n+1)\theta}{2}\right) \right) \]

### Step 2: Sum of sine terms
The sum inside the parentheses can be expressed as:

\[ S = \sin\left(\frac{3\theta}{2}\right) + \sin\left(\frac{5\theta}{2}\right) + \sin\left(\frac{7\theta}{2}\right) + \ldots + \sin\left(\frac{(2n+1)\theta}{2}\right) \]

This is an arithmetic series of sine functions. The general formula for the sum of sine functions can be used here:

\[ S = \frac{\sin\left(\frac{n \cdot d}{2}\right) \cdot \sin\left(\frac{a + l}{2}\right)}{\sin\left(\frac{d}{2}\right)} \]

where \( a \) is the first term, \( l \) is the last term, \( n \) is the number of terms, and \( d \) is the common difference.

### Step 3: Apply the formula
In our case:
- \( a = \frac{3\theta}{2} \)
- \( l = \frac{(2n+1)\theta}{2} \)
- \( d = \theta \)
- Number of terms \( n = n \)

Thus, we can write:

\[ S = \frac{\sin\left(\frac{n \cdot \theta}{2}\right) \cdot \sin\left(\frac{\frac{3\theta}{2} + \frac{(2n+1)\theta}{2}}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

### Step 4: Simplify the expression
Now, simplifying the sine terms:

\[ \frac{3\theta}{2} + \frac{(2n+1)\theta}{2} = \frac{(2n + 4)\theta}{2} \]

Thus:

\[ S = \frac{\sin\left(\frac{n \cdot \theta}{2}\right) \cdot \sin\left(\frac{(n + 2)\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

### Step 5: Substitute back into \( f_n(\theta) \)
Now substituting \( S \) back into \( f_n(\theta) \):

\[ f_n(\theta) = 2 \sin\left(\frac{\theta}{2}\right) \cdot \frac{\sin\left(\frac{n \cdot \theta}{2}\right) \cdot \sin\left(\frac{(n + 2)\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

This simplifies to:

\[ f_n(\theta) = 2 \sin\left(\frac{n \cdot \theta}{2}\right) \cdot \sin\left(\frac{(n + 2)\theta}{2}\right) \]

### Step 6: Evaluate specific cases
To find the values of \( f_n(\theta) \) for specific angles, we can substitute \( \theta \) with \( \frac{2\pi}{n} \) and other values as needed.

### Conclusion
After evaluating the function for specific angles, we can determine which options are correct based on the results obtained.