Find the value of \( 45^{\circ}-\sin ^{2} 30^{\circ} \) (a) \( \sin 45^{\circ} \cdot \cos 40^{\circ} \)

To find the value of \( 45^{\circ} - \sin^2 30^{\circ} \), we first need to calculate \( \sin^2 30^{\circ} \). We know that: \[ \sin 30^{\circ} = \frac{1}{2} \] Thus, \[ \sin^2 30^{\circ} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] Now we can substitute this value into the expression: \[ 45^{\circ} - \sin^2 30^{\circ} = 45^{\circ} - \frac{1}{4} \] Next, we need to convert \( 45^{\circ} \) into a fraction that can be combined with \( \frac{1}{4} \). We can express \( 45^{\circ} \) in terms of a fraction of degrees: \[ 45^{\circ} = \frac{45}{1} = \frac{180}{4} \] Now we can perform the subtraction: \[ \frac{180}{4} - \frac{1}{4} = \frac{180 - 1}{4} = \frac{179}{4} \] Next, we need to evaluate \( \sin 45^{\circ} \cdot \cos 40^{\circ} \) to see if it equals \( \frac{179}{4} \). We know: \[ \sin 45^{\circ} = \frac{\sqrt{2}}{2} \] Now we need to find \( \cos 40^{\circ} \). We will leave it as \( \cos 40^{\circ} \) for now, so we have: \[ \sin 45^{\circ} \cdot \cos 40^{\circ} = \frac{\sqrt{2}}{2} \cdot \cos 40^{\circ} \] To compare \( \frac{179}{4} \) with \( \frac{\sqrt{2}}{2} \cdot \cos 40^{\circ} \), we need to evaluate \( \frac{\sqrt{2}}{2} \cdot \cos 40^{\circ} \). However, without specific values for \( \cos 40^{\circ} \), we cannot directly compute this. Instead, we can conclude that \( 45^{\circ} - \sin^2 30^{\circ} = \frac{179}{4} \) is a numerical value, while \( \sin 45^{\circ} \cdot \cos 40^{\circ} \) is a product of trigonometric functions. Thus, the final answer is: \[ \boxed{\frac{179}{4}} \]