Find the value of
`cos^(2)1^(@)+cos^(2)2^(@)+cos^(2)3^(@)+ -----+cos^(2)87^(@)+cos^(2)88^(@)+cos^(2)89^(@)+cos^(2)90^(@)`

Options

A. `44(1)/(2)`
B. 44
C. `45(1)/(2)`
D. N.O.T.

Answer

To find the value of the expression \( \cos^2(1^\circ) + \cos^2(2^\circ) + \cos^2(3^\circ) + \ldots + \cos^2(90^\circ) \), we can use the properties of complementary angles in trigonometry.

### Step-by-Step Solution:

1. **Identify Complementary Angles**:
- Recall that \( \cos(90^\circ - \theta) = \sin(\theta) \). This means that for every angle \( \theta \), there is a complementary angle \( 90^\circ - \theta \).
- For example, \( \cos(1^\circ) = \sin(89^\circ) \), \( \cos(2^\circ) = \sin(88^\circ) \), and so on.

2. **Pairing the Terms**:
- We can pair the terms in the sum:
- \( \cos^2(1^\circ) + \cos^2(89^\circ) \)
- \( \cos^2(2^\circ) + \cos^2(88^\circ) \)
- \( \cos^2(3^\circ) + \cos^2(87^\circ) \)
- ...
- \( \cos^2(44^\circ) + \cos^2(46^\circ) \)
- The only unpaired terms are \( \cos^2(45^\circ) \) and \( \cos^2(90^\circ) \).

3. **Using the Identity**:
- For each pair \( \cos^2(k^\circ) + \cos^2(90^\circ - k^\circ) \):
\[ \cos^2(k^\circ) + \sin^2(k^\circ) = 1 \]
- Thus, each pair sums to 1.

4. **Counting the Pairs**:
- There are 44 pairs from \( k = 1 \) to \( k = 44 \).
- Therefore, the contribution from these pairs is \( 44 \times 1 = 44 \).

5. **Adding the Remaining Terms**:
- Now, we need to add \( \cos^2(45^\circ) \) and \( \cos^2(90^\circ) \):
- \( \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)
- \( \cos^2(90^\circ) = 0 \)

6. **Final Calculation**:
- The total sum is:
\[ 44 + \frac{1}{2} + 0 = 44.5 \]

### Conclusion:
The value of \( \cos^2(1^\circ) + \cos^2(2^\circ) + \ldots + \cos^2(90^\circ) \) is \( \boxed{44.5} \).