`sin ^(2) 25^(@) + sin ^(2) 65^(@) ` is equal to

Options

A. 0
B. 1
C. `1/2`
D. 2

Answer

To solve the problem \( \sin^2(25^\circ) + \sin^2(65^\circ) \), we can use the identity for sine and cosine. Here’s a step-by-step solution:

### Step 1: Recognize the relationship between angles
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
This means:
\[ \sin(65^\circ) = \cos(25^\circ) \]

### Step 2: Substitute the sine function
Using the identity from Step 1, we can rewrite \( \sin^2(65^\circ) \):
\[ \sin^2(65^\circ) = \cos^2(25^\circ) \]

### Step 3: Rewrite the original expression
Now, substitute \( \sin^2(65^\circ) \) into the original expression:
\[ \sin^2(25^\circ) + \sin^2(65^\circ) = \sin^2(25^\circ) + \cos^2(25^\circ) \]

### Step 4: Apply the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Thus, we can conclude:
\[ \sin^2(25^\circ) + \cos^2(25^\circ) = 1 \]

### Final Answer
Therefore, the value of \( \sin^2(25^\circ) + \sin^2(65^\circ) \) is:
\[ \boxed{1} \]