$$ \hat { \operatorname { 5 } } _ { \cos 75 ^ { \circ } \cdot \cos 45 ^ { \circ } - \sin 75 ^ { \circ } \cdot \sin 45 ^ { \circ } } $$

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To simplify the expression $$ \hat { \operatorname { 5 } } _ { \cos 75 ^ { \circ } \cdot \cos 45 ^ { \circ } - \sin 75 ^ { \circ } \cdot \sin 45 ^ { \circ } } $$, we can use the cosine of the sum of angles formula:

$$
\cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b
$$

In this case, we can identify $$ a = 75^\circ $$ and $$ b = 45^\circ $$. Therefore, we have:

$$
\cos(75^\circ + 45^\circ) = \cos 75^\circ \cdot \cos 45^\circ - \sin 75^\circ \cdot \sin 45^\circ
$$

Calculating $$ 75^\circ + 45^\circ $$:

$$
75^\circ + 45^\circ = 120^\circ
$$

Thus, we can rewrite the expression as:

$$
\cos 120^\circ
$$

Now, we know that:

$$
\cos 120^\circ = -\frac{1}{2}
$$

So, we can substitute this back into the original expression:

$$
\hat { \operatorname { 5 } } _ { -\frac{1}{2} }
$$

The final result is:

$$
\hat { \operatorname { 5 } } _ { -\frac{1}{2} }
$$
$$\hat { \operatorname { 5 } } _ { -\frac{1}{2} }$$

\( \hat { \operatorname { 5 } } _ { \cos 75 ^ { \circ } \cdot \cos 45 ^ { \circ } - \sin 75 ^ { \circ } \cdot \sin 45 ^ { \circ } } \)

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