`sin^(2)5^(@) + sin^(2)25^(@) + sin^(2)45^(@) + sin^(2) 65^(@) + sin^(2) 85^(@)` is equal to:

Options

A. 2.5
B. 3
C. 1.5
D. 2

Answer

To solve the expression \( \sin^2(5^\circ) + \sin^2(25^\circ) + \sin^2(45^\circ) + \sin^2(65^\circ) + \sin^2(85^\circ) \), we can use some properties of sine and trigonometric identities.

### Step 1: Use the identity \( \sin(90^\circ - x) = \cos(x) \)

We can pair the sine squares of angles that add up to \( 90^\circ \):
- \( \sin^2(5^\circ) \) and \( \sin^2(85^\circ) \)
- \( \sin^2(25^\circ) \) and \( \sin^2(65^\circ) \)

Using the identity, we have:
- \( \sin(85^\circ) = \cos(5^\circ) \)
- \( \sin(65^\circ) = \cos(25^\circ) \)

### Step 2: Rewrite the expression using the identity

Now, we can rewrite the pairs:
\[ \sin^2(5^\circ) + \sin^2(85^\circ) = \sin^2(5^\circ) + \cos^2(5^\circ) = 1 \]
\[ \sin^2(25^\circ) + \sin^2(65^\circ) = \sin^2(25^\circ) + \cos^2(25^\circ) = 1 \]

### Step 3: Calculate \( \sin^2(45^\circ) \)

We know that:
\[ \sin^2(45^\circ) = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} \]

### Step 4: Combine all the results

Now, we can combine everything:
\[ \sin^2(5^\circ) + \sin^2(85^\circ) + \sin^2(25^\circ) + \sin^2(65^\circ) + \sin^2(45^\circ) = 1 + 1 + \frac{1}{2} \]
\[ = 2 + \frac{1}{2} = 2.5 \]

### Final Answer

Thus, the value of \( \sin^2(5^\circ) + \sin^2(25^\circ) + \sin^2(45^\circ) + \sin^2(65^\circ) + \sin^2(85^\circ) \) is \( \frac{5}{2} \) or \( 2.5 \).

---