The value of `sin^2 5^@ + sin^2 10^@ + sin^2 15^@ + sin^2 20^@ + ... + sin^2 90^@` is:

Options

A. 7
B. 8
C. 9
D. `(19)/(2)`

Answer

To find the value of \( \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \sin^2 20^\circ + \ldots + \sin^2 90^\circ \), we can use the properties of sine and cosine functions.

### Step-by-Step Solution:

1. **Identify the Series**:
The series we need to evaluate is:
\[ S = \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \ldots + \sin^2 90^\circ \]

2. **Pair the Terms**:
Notice that \( \sin^2 (90^\circ - x) = \cos^2 x \). We can pair terms from the beginning and the end of the series:
\[ \sin^2 5^\circ + \sin^2 85^\circ, \quad \sin^2 10^\circ + \sin^2 80^\circ, \quad \sin^2 15^\circ + \sin^2 75^\circ, \quad \ldots, \quad \sin^2 45^\circ + \sin^2 45^\circ \]

3. **Use the Identity**:
Each pair can be expressed using the identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Therefore, we can write:
- \( \sin^2 5^\circ + \sin^2 85^\circ = 1 \)
- \( \sin^2 10^\circ + \sin^2 80^\circ = 1 \)
- \( \sin^2 15^\circ + \sin^2 75^\circ = 1 \)
- \( \sin^2 20^\circ + \sin^2 70^\circ = 1 \)
- \( \sin^2 25^\circ + \sin^2 65^\circ = 1 \)
- \( \sin^2 30^\circ + \sin^2 60^\circ = 1 \)
- \( \sin^2 35^\circ + \sin^2 55^\circ = 1 \)
- \( \sin^2 40^\circ + \sin^2 50^\circ = 1 \)

4. **Count the Pairs**:
There are 8 pairs from \( \sin^2 5^\circ \) to \( \sin^2 85^\circ \), each summing to 1:
\[ 8 \times 1 = 8 \]

5. **Add the Remaining Term**:
The only term left is \( \sin^2 45^\circ \):
\[ \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

6. **Final Calculation**:
Now, add the sum of the pairs and the remaining term:
\[ S = 8 + \frac{1}{2} = \frac{16}{2} + \frac{1}{2} = \frac{17}{2} \]

### Conclusion:
The value of \( \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \ldots + \sin^2 90^\circ \) is:
\[ \boxed{\frac{17}{2}} \]