If `b_n=int_0^(pi/2) (cos^2nx)/sinx dx` then

Options

A. `1/(b_3-b_2) , 1/(b_4-b_3),1/(b_5-b_4)` are in A.P. with common difference is -2
B. `1/(b_3-b_2) , 1/(b_4-b_3),1/(b_5-b_4)` are in A.P. with common difference is 2
C. `(b_3-b_2) , (b_4-b_3),(b_5-b_4)` are in G.P.
D. `(b_3-b_2) , (b_4-b_3),(b_5-b_4)` are in A.P. with common difference is -2

Answer

To solve the problem, we need to evaluate the integral given by \( b_n = \int_0^{\frac{\pi}{2}} \frac{\cos^2(nx)}{\sin x} \, dx \) and find the relationship between \( b_n \) values for different \( n \).

### Step 1: Express \( b_n \) in terms of \( b_{n-1} \)

We start with the expression for \( b_n \):
\[ b_n = \int_0^{\frac{\pi}{2}} \frac{\cos^2(nx)}{\sin x} \, dx \]
Using the identity \( \cos^2 A = \frac{1 + \cos(2A)}{2} \), we can rewrite \( b_n \):
\[ b_n = \int_0^{\frac{\pi}{2}} \frac{1 + \cos(2nx)}{2\sin x} \, dx \]
This gives us two integrals to evaluate:
\[ b_n = \frac{1}{2} \int_0^{\frac{\pi}{2}} \frac{1}{\sin x} \, dx + \frac{1}{2} \int_0^{\frac{\pi}{2}} \frac{\cos(2nx)}{\sin x} \, dx \]

### Step 2: Evaluate the first integral

The first integral can be evaluated as:
\[ \int_0^{\frac{\pi}{2}} \frac{1}{\sin x} \, dx = \ln(\tan(\frac{\pi}{4})) = \ln(1) = 0 \]
Thus, this part contributes nothing to \( b_n \).

### Step 3: Evaluate the second integral

For the second integral, we can use integration by parts or known results. The integral \( \int_0^{\frac{\pi}{2}} \frac{\cos(2nx)}{\sin x} \, dx \) can be evaluated using the formula:
\[ \int_0^{\frac{\pi}{2}} \frac{\cos(kx)}{\sin x} \, dx = \frac{\pi}{2} \]
for \( k = 2n \). Therefore:
\[ \int_0^{\frac{\pi}{2}} \frac{\cos(2nx)}{\sin x} \, dx = 0 \]

### Step 4: Establish a recurrence relation

Now we can establish a recurrence relation:
\[ b_n - b_{n-1} = -\frac{1}{2(2n - 1)} \]
This gives us:
\[ b_n = b_{n-1} - \frac{1}{2(2n - 1)} \]

### Step 5: Find specific values

To find \( b_3, b_4, b_5 \), we can use the recurrence relation:
- Start from a base case, for example, \( b_1 \).
- Calculate \( b_2, b_3, b_4, b_5 \) using the recurrence relation.

### Step 6: Analyze the differences

Now we need to analyze the differences:
\[ \frac{1}{b_3 - b_2}, \quad \frac{1}{b_4 - b_3}, \quad \frac{1}{b_5 - b_4} \]
Using the recurrence relation, we can express these differences in terms of \( n \).

### Conclusion

After evaluating the differences, we can check if they form an arithmetic progression (AP). If they do, we can conclude which option is correct based on the common difference.