What is `sin^2 50^@ + sin^2 40^@` equal to?

Options

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

Answer

To solve the problem \( \sin^2 50^\circ + \sin^2 40^\circ \), we can use the identity involving sine and cosine functions. Here’s a step-by-step solution:

### Step 1: Rewrite \( \sin 40^\circ \)
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
Thus, we can express \( \sin 40^\circ \) in terms of \( \sin 50^\circ \):
\[ \sin 40^\circ = \sin(90^\circ - 50^\circ) = \cos 50^\circ \]

### Step 2: Substitute \( \sin 40^\circ \) into the equation
Now we can substitute \( \sin 40^\circ \) in our original expression:
\[ \sin^2 50^\circ + \sin^2 40^\circ = \sin^2 50^\circ + \cos^2 50^\circ \]

### Step 3: Apply the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Applying this identity for \( \theta = 50^\circ \):
\[ \sin^2 50^\circ + \cos^2 50^\circ = 1 \]

### Conclusion
Therefore, we find that:
\[ \sin^2 50^\circ + \sin^2 40^\circ = 1 \]

### Final Answer
The value of \( \sin^2 50^\circ + \sin^2 40^\circ \) is \( 1 \).

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