If `f(x)=1/pi int_0^(pi/2) sin^2(ntheta)/sin^2theta d(theta)` then evaluate `(f(15)+f(3))/(f(15)-f(9))`

Answer

To solve the problem, we need to evaluate the expression \((f(15) + f(3)) / (f(15) - f(9))\) where \(f(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(n\theta)}{\sin^2(\theta)} d\theta\).

### Step 1: Determine \(f(n)\)
From the video transcript, we can derive that:
\[ f(n) = \frac{n}{2} \]
This means we can find \(f(15)\), \(f(3)\), and \(f(9)\) directly.

### Step 2: Calculate \(f(15)\)
Using the formula:
\[ f(15) = \frac{15}{2} = 7.5 \]

### Step 3: Calculate \(f(3)\)
Using the formula:
\[ f(3) = \frac{3}{2} = 1.5 \]

### Step 4: Calculate \(f(9)\)
Using the formula:
\[ f(9) = \frac{9}{2} = 4.5 \]

### Step 5: Substitute values into the expression
Now we substitute \(f(15)\), \(f(3)\), and \(f(9)\) into the expression:
\[ \frac{f(15) + f(3)}{f(15) - f(9)} = \frac{7.5 + 1.5}{7.5 - 4.5} \]

### Step 6: Simplify the numerator and denominator
Calculating the numerator:
\[ 7.5 + 1.5 = 9 \]
Calculating the denominator:
\[ 7.5 - 4.5 = 3 \]

### Step 7: Final calculation
Now we can compute the final result:
\[ \frac{9}{3} = 3 \]

Thus, the final answer is:
\[ \boxed{3} \]