Without using tables or calculator, evaluate and simplify \[ \frac{\sin 30^{\circ}+\sin 45^{\circ}}{\cos 60^{\circ}-1} \]

To start simplifying the expression \[ \frac{\sin 30^{\circ}+\sin 45^{\circ}}{\cos 60^{\circ}-1}, \] we first need to evaluate the trigonometric functions involved. Recall that: - \(\sin 30^{\circ} = \frac{1}{2}\) - \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) - \(\cos 60^{\circ} = \frac{1}{2}\) Now we can substitute these values into the expression. The numerator becomes: \[ \sin 30^{\circ} + \sin 45^{\circ} = \frac{1}{2} + \frac{\sqrt{2}}{2} = \frac{1 + \sqrt{2}}{2}. \] The denominator becomes: \[ \cos 60^{\circ} - 1 = \frac{1}{2} - 1 = -\frac{1}{2}. \] Now, we can put these results together: \[ \frac{\sin 30^{\circ} + \sin 45^{\circ}}{\cos 60^{\circ} - 1} = \frac{\frac{1 + \sqrt{2}}{2}}{-\frac{1}{2}}. \] To simplify, we multiply by the reciprocal of the denominator: \[ = \frac{1 + \sqrt{2}}{2} \cdot \left(-2\right) = -(1 + \sqrt{2}). \] Thus, the final simplified expression is: \[ \boxed{-(1 + \sqrt{2})}. \]