The value of \(\left[\frac{\sin^{2}25^\circ + \sin^{2}65^\circ}{\cos^{2}25^\circ + \cos^{2}65^\circ} + \sin^{2}57^\circ + \cos 57^\circ \sin 33^\circ\right]\) is:
2
25° + 65° = 90°, so sin 65° = cos 25° and cos 65° = sin 25°.
Numerator: sin225° + cos225° = 1; denominator: cos225° + sin225° = 1. So the fraction is 1.
57° + 33° = 90°, so sin 33° = cos 57°, and sin257° + cos 57° sin 33° = sin257° + cos257° = 1.
Value = 1 + 1 = 2.
Hence, the answer is 2.
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