`sin^(2)21^(@) + sin^(2) 69^(@)` is equal to

Options

A. `2sin^(2)21^(@)`
B. `2sin^(2)69^(@)`
C. 1
D. 0

Answer

To solve the expression \( \sin^2(21^\circ) + \sin^2(69^\circ) \), we can use the trigonometric identity that relates sine and cosine. Here are the steps:

### Step-by-Step Solution:

1. **Write the expression**:
\[ \sin^2(21^\circ) + \sin^2(69^\circ) \]

2. **Use the complementary angle identity**:
We know that \( \sin(90^\circ - \theta) = \cos(\theta) \). Therefore, we can express \( \sin(69^\circ) \) as:
\[ \sin(69^\circ) = \sin(90^\circ - 21^\circ) = \cos(21^\circ) \]

3. **Substitute \( \sin(69^\circ) \)**:
Now, we can rewrite \( \sin^2(69^\circ) \) using the identity:
\[ \sin^2(69^\circ) = \cos^2(21^\circ) \]
So, the expression becomes:
\[ \sin^2(21^\circ) + \cos^2(21^\circ) \]

4. **Apply the Pythagorean identity**:
We know from the Pythagorean identity that:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Therefore:
\[ \sin^2(21^\circ) + \cos^2(21^\circ) = 1 \]

5. **Final result**:
Thus, we conclude that:
\[ \sin^2(21^\circ) + \sin^2(69^\circ) = 1 \]

### Conclusion:
The value of \( \sin^2(21^\circ) + \sin^2(69^\circ) \) is equal to \( 1 \).

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