The value of `sin^2 1^@ + sin^2 3^@ + sin^2 5^@ +…+ sin^2 87^@ + sin^2 89^@` is

Options

A. 22
B. `22(1)/2`
C. 23
D. `22(1)/4`

Answer

To solve the problem, we need to find the value of the sum:

\[ S = \sin^2 1^\circ + \sin^2 3^\circ + \sin^2 5^\circ + \ldots + \sin^2 87^\circ + \sin^2 89^\circ \]

### Step 1: Identify the Pattern
The angles in the sum are all odd degrees from 1 to 89. There are a total of 45 terms in this sequence (1, 3, 5, ..., 89).

### Step 2: Use the Identity
We can use the identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

This means that for each angle \( \theta \), we can pair \( \sin^2 \theta \) with \( \sin^2 (90^\circ - \theta) \), which is equal to \( \cos^2 \theta \).

### Step 3: Pair the Terms
We can pair the terms in the sum:

- \( \sin^2 1^\circ + \sin^2 89^\circ = \sin^2 1^\circ + \cos^2 1^\circ = 1 \)
- \( \sin^2 3^\circ + \sin^2 87^\circ = \sin^2 3^\circ + \cos^2 3^\circ = 1 \)
- \( \sin^2 5^\circ + \sin^2 85^\circ = \sin^2 5^\circ + \cos^2 5^\circ = 1 \)
- ...
- \( \sin^2 43^\circ + \sin^2 47^\circ = \sin^2 43^\circ + \cos^2 43^\circ = 1 \)
- Finally, \( \sin^2 45^\circ = \frac{1}{2} \)

### Step 4: Count the Pairs
There are 22 pairs (from \( 1^\circ \) to \( 43^\circ \)), each contributing 1 to the total sum, plus the single term \( \sin^2 45^\circ \).

### Step 5: Calculate the Total
The total sum can be calculated as follows:

\[ S = 22 \times 1 + \frac{1}{2} = 22 + \frac{1}{2} = 22.5 \]

Thus, the value of the sum \( S \) is:

\[ \boxed{22.5} \]