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

To simplify the expression \( \sin ^{2} 108^{\circ}+\sin ^{2} 18^{\circ} \), we can use the fact that \( \sin(108^{\circ}) = \sin(180^{\circ} - 72^{\circ}) = \sin(72^{\circ}) \). Thus: \[ \sin^{2}(108^{\circ}) = \sin^{2}(72^{\circ}) \] Now, we can simplify the expression as: \[ \sin^{2}(108^{\circ}) + \sin^{2}(18^{\circ}) = \sin^{2}(72^{\circ}) + \sin^{2}(18^{\circ) \] Using the identity \( \sin^{2}(x) + \sin^{2}(y) = 1 - \frac{1}{2} \cos(2x) - \frac{1}{2} \cos(2y) \) might be complex but let's look for a direct pattern. Recall the identity: \[ \sin^{2}(x) + \cos^{2}(x) = 1 \] To further connect these, notice that \( 108 + 18 = 126 \) is not directly helpful. However, from the sine angle addition identity, recognize that both angles are related through the properties of sine in terms of each other. Instead, we can also apply the Pythagorean identity: \[ \sin^{2}(x) + \cos^{2}(x) = 1 \] Finally, using known sine values \( \sin(18^{\circ}) \) and \( \sin(72^{\circ}) = \cos(18^{\circ}) \): We know, \[ \sin(72^{\circ}) = \cos(18^{\circ}) \text{ and thus } \cos^{2}(18^{\circ}) + \sin^{2}(18^{\circ}) = 1 \] Putting it together beautifully simplifies our expression finally to: \[ \sin^{2}(108^{\circ})+\sin^{2}(18^{\circ}) = 1 \] Thus, the final simplified result is \( 1 \).