If `f (n)= 1/pi int _(0) ^(pi//2) (sin ^(2) (n theta) d theta)/(sin ^(2) theta), n in N,` then evulate `(f (15)+ f (3))/( f (12) -f (10)).`

Answer

To solve the problem, we need to evaluate the expression \((f(15) + f(3)) / (f(12) - f(10))\) where \(f(n) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(n\theta)}{\sin^2(\theta)} d\theta\).

### Step-by-Step Solution:

1. **Identify the Function**:
We have the function defined as:
\[ f(n) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(n\theta)}{\sin^2(\theta)} d\theta \]

2. **Finding a Pattern**:
From the calculations in the video, we observe that:
\[ f(n) = \frac{n}{2} \]
This suggests that for any natural number \(n\), \(f(n)\) can be expressed as \(n/2\).

3. **Calculate \(f(15)\)**:
Using the pattern:
\[ f(15) = \frac{15}{2} = 7.5 \]

4. **Calculate \(f(3)\)**:
Using the pattern:
\[ f(3) = \frac{3}{2} = 1.5 \]

5. **Calculate \(f(12)\)**:
Using the pattern:
\[ f(12) = \frac{12}{2} = 6 \]

6. **Calculate \(f(10)\)**:
Using the pattern:
\[ f(10) = \frac{10}{2} = 5 \]

7. **Substituting Values into the Expression**:
Now we substitute the values into the expression:
\[ \frac{f(15) + f(3)}{f(12) - f(10)} = \frac{7.5 + 1.5}{6 - 5} \]

8. **Simplifying the Expression**:
Calculate the numerator and denominator:
\[ = \frac{9}{1} = 9 \]

### Final Answer:
\[ \text{The value of } \frac{f(15) + f(3)}{f(12) - f(10)} \text{ is } 9. \]