\( | \left. \begin{array} { l l } { 3.7 } & { \cos ^ { 2 } 45 ^ { \circ } - 1 } \\ { 3.8 } & { 2 \cos ^ { 2 } 45 ^ { \circ } - 1 } \end{array} \right. \)

To solve the given expressions, we need to evaluate \( \cos^2 45^\circ \) first. 1. **Calculate \( \cos 45^\circ \)**: \[ \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \] 2. **Calculate \( \cos^2 45^\circ \)**: \[ \cos^2 45^\circ = \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} = \frac{1}{2} \] Now we can substitute \( \cos^2 45^\circ \) into the two expressions. ### For the first expression: \[ 3.7: \quad \cos^2 45^\circ - 1 = \frac{1}{2} - 1 = \frac{1}{2} - \frac{2}{2} = -\frac{1}{2} \] ### For the second expression: \[ 3.8: \quad 2 \cos^2 45^\circ - 1 = 2 \cdot \frac{1}{2} - 1 = 1 - 1 = 0 \] ### Final Results: - For \( 3.7 \): \( -\frac{1}{2} \) - For \( 3.8 \): \( 0 \) Thus, the final results are: \[ \left. \begin{array} { l l } { 3.7 } & { -\frac{1}{2} } \\ { 3.8 } & { 0 } \end{array} \right. \]