\( \frac{\text { Simplify: }}{\sin ^{2} 108+\sin ^{2} 18} \)

To simplify the expression: \[ \sin^2 108^\circ + \sin^2 18^\circ \] we can use trigonometric identities and symmetry properties. ### Step 1: Utilize Angle Relationships Notice that: \[ \sin 108^\circ = \sin (180^\circ - 72^\circ) = \sin 72^\circ \] So, the expression becomes: \[ \sin^2 72^\circ + \sin^2 18^\circ \] ### Step 2: Apply the Pythagorean Identity for Sine Squared Recall the identity: \[ \sin^2 A + \sin^2 B = 1 - \frac{\cos(2A) + \cos(2B)}{2} \] Applying this to our expression: \[ \sin^2 72^\circ + \sin^2 18^\circ = 1 - \frac{\cos(144^\circ) + \cos(36^\circ)}{2} \] ### Step 3: Evaluate the Cosines We know that: \[ \cos 144^\circ = \cos (180^\circ - 36^\circ) = -\cos 36^\circ \] Substituting back: \[ \cos 144^\circ + \cos 36^\circ = -\cos 36^\circ + \cos 36^\circ = 0 \] So, the expression simplifies to: \[ \sin^2 72^\circ + \sin^2 18^\circ = 1 - \frac{0}{2} = 1 \] ### Conclusion The simplified form of the expression is: \[ \boxed{1} \]