Find the value of `(sin^2 33^@ + sin^2 57^@)`

Answer

To find the value of \( \sin^2 33^\circ + \sin^2 57^\circ \), we can use the trigonometric identity that relates sine and cosine.

### Step 1: Rewrite \( \sin^2 57^\circ \)
Using the identity \( \sin(90^\circ - \theta) = \cos(\theta) \), we can express \( \sin^2 57^\circ \) in terms of cosine:
\[ \sin^2 57^\circ = \sin^2(90^\circ - 33^\circ) = \cos^2 33^\circ \]

### Step 2: Substitute into the equation
Now we can substitute this back into our original expression:
\[ \sin^2 33^\circ + \sin^2 57^\circ = \sin^2 33^\circ + \cos^2 33^\circ \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Applying this identity for \( \theta = 33^\circ \):
\[ \sin^2 33^\circ + \cos^2 33^\circ = 1 \]

### Final Answer
Thus, the value of \( \sin^2 33^\circ + \sin^2 57^\circ \) is:
\[ \boxed{1} \]