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

Calculate the value by following steps: - step0: Calculate: \(\sin\left(108^{2}\right)+\sin\left(18^{2}\right)\) - step1: Transform the expression: \(2\sin\left(5994\right)\cos\left(5670\right)\) - step2: Transform the expression: \(\sin\left(11664\right)+\sin\left(324\right)\) To simplify the expression \( \sin^{2} 108^{\circ} + \sin^{2} 18^{\circ} \), we can use the following steps: 1. **Recognize the angles**: We know that \( 108^{\circ} \) and \( 18^{\circ} \) are related. Specifically, \( 108^{\circ} = 90^{\circ} + 18^{\circ} \). 2. **Use the sine identity**: The sine of an angle can be expressed in terms of its complementary angle. Thus, we can use the identity: \[ \sin(90^{\circ} + x) = \cos(x) \] Therefore, \( \sin(108^{\circ}) = \cos(18^{\circ}) \). 3. **Substituting the identity**: We can rewrite the expression: \[ \sin^{2}(108^{\circ}) = \cos^{2}(18^{\circ}) \] So, the expression becomes: \[ \cos^{2}(18^{\circ}) + \sin^{2}(18^{\circ}) \] 4. **Apply the Pythagorean identity**: According to the Pythagorean identity, we have: \[ \cos^{2}(x) + \sin^{2}(x) = 1 \] Therefore: \[ \cos^{2}(18^{\circ}) + \sin^{2}(18^{\circ}) = 1 \] Thus, the simplified result of \( \sin^{2} 108^{\circ} + \sin^{2} 18^{\circ} \) is \( 1 \).