+ Simplify \( \frac{\sin 150^{\circ}+\cos ^{2} x-1}{2} \)

To simplify the expression \( \frac{\sin 150^{\circ}+\cos ^{2} x-1}{2} \), we start by calculating \( \sin 150^{\circ} \). We know that \( \sin 150^{\circ} = \sin (180^{\circ} - 30^{\circ}) = \sin 30^{\circ} = \frac{1}{2} \). Now, we can substitute this value into the expression: \[ \frac{\sin 150^{\circ} + \cos^2 x - 1}{2} = \frac{\frac{1}{2} + \cos^2 x - 1}{2} \] Next, we simplify the numerator: \[ \frac{1}{2} - 1 + \cos^2 x = \cos^2 x - \frac{1}{2} \] Putting it all together, we have: \[ \frac{\cos^2 x - \frac{1}{2}}{2} \] Thus, the expression simplifies to: \[ \frac{\cos^2 x - \frac{1}{2}}{2} \] This is the simplified form of the given expression.