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 evaluate the expression \((f(15) + f(3)) / (f(12) - f(10))\), we first need to determine the function \(f(n)\) defined as:

\[ f(n) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(n\theta)}{\sin^2(\theta)} d\theta \]

### Step 1: Find \(f(1)\)

For \(n = 1\):

\[ f(1) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(\theta)}{\sin^2(\theta)} d\theta = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} 1 \, d\theta \]

Calculating the integral:

\[ = \frac{1}{\pi} \left[ \theta \right]_0^{\frac{\pi}{2}} = \frac{1}{\pi} \left( \frac{\pi}{2} - 0 \right) = \frac{1}{2} \]

### Step 2: Find \(f(2)\)

For \(n = 2\):

\[ f(2) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(2\theta)}{\sin^2(\theta)} d\theta \]

Using the identity \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\):

\[ = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{4\sin^2(\theta)\cos^2(\theta)}{\sin^2(\theta)} d\theta = \frac{4}{\pi} \int_0^{\frac{\pi}{2}} \cos^2(\theta) d\theta \]

Using the identity \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\):

\[ = \frac{4}{\pi} \int_0^{\frac{\pi}{2}} \frac{1 + \cos(2\theta)}{2} d\theta = \frac{2}{\pi} \left[ \theta + \frac{\sin(2\theta)}{2} \right]_0^{\frac{\pi}{2}} = \frac{2}{\pi} \left( \frac{\pi}{2} + 0 \right) = 1 \]

### Step 3: Find \(f(3)\)

For \(n = 3\):

\[ f(3) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{\sin^2(3\theta)}{\sin^2(\theta)} d\theta \]

Using the identity \(\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\):

\[ = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \frac{(3\sin(\theta) - 4\sin^3(\theta))^2}{\sin^2(\theta)} d\theta \]

After simplification, we find:

\[ = \frac{3}{2} \]

### Step 4: Identify the Pattern

From the calculations:

- \(f(1) = \frac{1}{2}\)
- \(f(2) = 1\)
- \(f(3) = \frac{3}{2}\)

We can see a pattern:

\[ f(n) = \frac{n}{2} \]

### Step 5: Calculate \(f(15)\), \(f(12)\), and \(f(10)\)

Using the pattern:

- \(f(15) = \frac{15}{2}\)
- \(f(12) = \frac{12}{2} = 6\)
- \(f(10) = \frac{10}{2} = 5\)

### Step 6: Substitute into the Expression

Now substituting these values into the expression:

\[ \frac{f(15) + f(3)}{f(12) - f(10)} = \frac{\frac{15}{2} + \frac{3}{2}}{6 - 5} = \frac{\frac{18}{2}}{1} = 9 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{9} \]