`sin^(2)5^(@)+sin^(2)10^(@)+sin^(2)15^(@)+ .....+sin^(2)85^(@)+sin^(2)90^(@)` is equal to

Options

A. `7(1)/(2)`
B. `8(1)/(2)`
C. 9
D. `9(1)/(2)`

Answer

To solve the problem \( \sin^2(5^\circ) + \sin^2(10^\circ) + \sin^2(15^\circ) + \ldots + \sin^2(85^\circ) + \sin^2(90^\circ) \), we can use the properties of complementary angles in trigonometry.

### Step-by-Step Solution:

1. **Identify the Terms**: The series consists of sine squares of angles from \( 5^\circ \) to \( 90^\circ \) in increments of \( 5^\circ \).

2. **Pair the Complementary Angles**: Notice that:
- \( \sin^2(5^\circ) \) pairs with \( \sin^2(85^\circ) \) (since \( 5^\circ + 85^\circ = 90^\circ \))
- \( \sin^2(10^\circ) \) pairs with \( \sin^2(80^\circ) \)
- \( \sin^2(15^\circ) \) pairs with \( \sin^2(75^\circ) \)
- \( \ldots \)
- \( \sin^2(40^\circ) \) pairs with \( \sin^2(50^\circ) \)
- \( \sin^2(45^\circ) \) stands alone since it is equal to \( \sin^2(45^\circ) \).

3. **Use the Identity**: For complementary angles, we have:
\[ \sin^2(\theta) + \sin^2(90^\circ - \theta) = 1 \]
Therefore, each pair sums to 1:
- \( \sin^2(5^\circ) + \sin^2(85^\circ) = 1 \)
- \( \sin^2(10^\circ) + \sin^2(80^\circ) = 1 \)
- \( \sin^2(15^\circ) + \sin^2(75^\circ) = 1 \)
- \( \sin^2(20^\circ) + \sin^2(70^\circ) = 1 \)
- \( \sin^2(25^\circ) + \sin^2(65^\circ) = 1 \)
- \( \sin^2(30^\circ) + \sin^2(60^\circ) = 1 \)
- \( \sin^2(35^\circ) + \sin^2(55^\circ) = 1 \)
- \( \sin^2(40^\circ) + \sin^2(50^\circ) = 1 \)

4. **Count the Pairs**: There are 8 pairs of complementary angles, each summing to 1. Therefore, the contribution from these pairs is:
\[ 8 \times 1 = 8 \]

5. **Add the Remaining Terms**: Now, we need to add the values of \( \sin^2(45^\circ) \) and \( \sin^2(90^\circ) \):
- \( \sin^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)
- \( \sin^2(90^\circ) = 1^2 = 1 \)

6. **Final Calculation**: Now, we sum everything:
\[ \text{Total} = 8 + \frac{1}{2} + 1 = 8 + 0.5 + 1 = 9.5 \]

### Conclusion:
Thus, the value of the series \( \sin^2(5^\circ) + \sin^2(10^\circ) + \sin^2(15^\circ) + \ldots + \sin^2(85^\circ) + \sin^2(90^\circ) \) is equal to \( 9.5 \).