What is `sin^(2)20^(@)+sin^(2)70^(@)` equal to ?

Options

A. 1
B. 0
C. `-1`
D. `(1)/(2)`

Explanation

To solve the expression \( \sin^2(20^\circ) + \sin^2(70^\circ) \), we can use the property of sine that relates angles that sum to \( 90^\circ \).

### Step-by-Step Solution:

1. **Identify the relationship between the angles**:
\[ 70^\circ = 90^\circ - 20^\circ \]
This means that \( \sin(70^\circ) = \cos(20^\circ) \).

2. **Use the sine identity**:
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
Therefore, we can write:
\[ \sin(70^\circ) = \cos(20^\circ) \]

3. **Substitute into the expression**:
Now, we can rewrite \( \sin^2(70^\circ) \):
\[ \sin^2(70^\circ) = \cos^2(20^\circ) \]

4. **Use the Pythagorean identity**:
Recall that:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Applying this to our angles:
\[ \sin^2(20^\circ) + \cos^2(20^\circ) = 1 \]

5. **Combine the results**:
Therefore, we can write:
\[ \sin^2(20^\circ) + \sin^2(70^\circ) = \sin^2(20^\circ) + \cos^2(20^\circ) = 1 \]

### Final Answer:
\[ \sin^2(20^\circ) + \sin^2(70^\circ) = 1 \]