If `rArrI_(n)= int_(a)^(a+pi//2)(cos^(2)nx)/(sinx) dx, "then" I_(2)-I_(1),I_(3)-I_(2),I_(4)-I_(3)` are in

Options

A. G.P.
B. A.P.
C. H.P.
D. none of these

Explanation

To solve the problem, we need to evaluate the differences \( I_2 - I_1 \), \( I_3 - I_2 \), and \( I_4 - I_3 \) for the integral

\[ I_n = \int_a^{a + \frac{\pi}{2}} \frac{\cos^2(nx)}{\sin x} \, dx \]

and determine if these differences are in a geometric progression (GP), harmonic progression (HP), or neither.

### Step 1: Expressing \( I_n \)

We start with the integral:

\[ I_n = \int_a^{a + \frac{\pi}{2}} \frac{\cos^2(nx)}{\sin x} \, dx \]

Using the periodicity of sine and cosine functions, we can rewrite the integral over a more convenient interval:

\[ I_n = \int_0^{\frac{\pi}{2}} \frac{\cos^2(nx)}{\sin x} \, dx \]

### Step 2: Finding the Differences

We need to find the differences \( I_2 - I_1 \), \( I_3 - I_2 \), and \( I_4 - I_3 \).

Using the identity for the difference of integrals:

\[ I_n - I_{n-1} = \int_0^{\frac{\pi}{2}} \left( \frac{\cos^2(nx)}{\sin x} - \frac{\cos^2((n-1)x)}{\sin x} \right) dx \]

This can be simplified as:

\[ I_n - I_{n-1} = \int_0^{\frac{\pi}{2}} \frac{\cos^2(nx) - \cos^2((n-1)x)}{\sin x} \, dx \]

### Step 3: Using the Cosine Square Identity

Using the identity \( \cos^2 A - \cos^2 B = (\cos A - \cos B)(\cos A + \cos B) \):

\[ \cos^2(nx) - \cos^2((n-1)x) = \left( \cos(nx) - \cos((n-1)x) \right) \left( \cos(nx) + \cos((n-1)x) \right) \]

### Step 4: Evaluating the Integral

The integral can be evaluated using the sine difference formula:

\[ I_n - I_{n-1} = \int_0^{\frac{\pi}{2}} \frac{\sin((2n-1)x)}{\sin x} \, dx \]

This integral can be computed to yield:

\[ I_n - I_{n-1} = -\frac{1}{2n - 1} \]

### Step 5: Finding Each Difference

Now we can compute:

1. \( I_2 - I_1 = -\frac{1}{2 \cdot 2 - 1} = -\frac{1}{3} \)
2. \( I_3 - I_2 = -\frac{1}{2 \cdot 3 - 1} = -\frac{1}{5} \)
3. \( I_4 - I_3 = -\frac{1}{2 \cdot 4 - 1} = -\frac{1}{7} \)

### Step 6: Checking for HP

To check if these differences are in HP, we need to verify the condition:

\[ \frac{2}{I_3 - I_2} = \frac{1}{I_2 - I_1} + \frac{1}{I_4 - I_3} \]

Substituting the values:

\[ \frac{2}{-\frac{1}{5}} = \frac{1}{-\frac{1}{3}} + \frac{1}{-\frac{1}{7}} \]

Calculating both sides:

Left side:

\[ -10 = -3 + -7 \]

Both sides equal, confirming that the differences are in HP.

### Conclusion

Thus, we conclude that \( I_2 - I_1 \), \( I_3 - I_2 \), and \( I_4 - I_3 \) are in harmonic progression (HP).