Simplify the following expressions: a) \( \sin ^{2} 108^{\circ}+\sin ^{2} 18^{\circ} \)

To simplify \( \sin ^{2} 108^{\circ}+\sin ^{2} 18^{\circ} \), we can use the identity that relates the sine of complementary angles. Notably, \( \sin 108^{\circ} = \sin(90^{\circ} - 18^{\circ}) = \cos 18^{\circ} \). Now we proceed with the simplification: 1. Calculate \( \sin^2 108^{\circ} = \cos^2 18^{\circ} \). 2. Therefore, the expression becomes \( \cos^2 18^{\circ} + \sin^2 18^{\circ} \). By employing the Pythagorean identity, we know that: \[ \cos^2 \theta + \sin^2 \theta = 1 \] Thus, we find: \[ \sin^2 108^{\circ} + \sin^2 18^{\circ} = 1 \] So the simplified expression is \( 1 \).