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

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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:
$$
\boxed{\frac{179}{4}}
$$
The value of $$ 45^{\circ} - \sin^2 30^{\circ} $$ is $$ \frac{179}{4} $$.

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

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