\( \left. \begin{array} { | l l } { 5.2 .5 } & { \frac { \cos ( \alpha - 90 ^ { \circ } ) \cdot \tan ( - \alpha ) } { \sin ( - \alpha ) \cdot \tan ( 720 ^ { \circ } - \alpha ) } } \\ { 5.2 .6 } & { \frac { \sin ( \beta - 180 ^ { \circ } ) \cdot \tan ( - \beta - 180 ^ { \circ } ) \cdot \cos ( 180 ^ { \circ } + \beta ) } { \cos ( - \beta ) \cdot \sin ( 360 ^ { \circ } + \beta ) } } \end{array} \right. \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\sin\left(\beta -180^{\circ}\right)\tan\left(-\beta -180^{\circ}\right)\cos\left(180^{\circ}+\beta \right)}{\cos\left(-\beta \right)\sin\left(360^{\circ}+\beta \right)}\) - step1: Rewrite the expression: \(\frac{-\sin\left(\beta \right)\tan\left(-\beta -180^{\circ}\right)\cos\left(180^{\circ}+\beta \right)}{\cos\left(-\beta \right)\sin\left(360^{\circ}+\beta \right)}\) - step2: Rewrite the expression: \(\frac{-\sin\left(\beta \right)\left(-\tan\left(\beta \right)\right)\cos\left(180^{\circ}+\beta \right)}{\cos\left(-\beta \right)\sin\left(360^{\circ}+\beta \right)}\) - step3: Rewrite the expression: \(\frac{-\sin\left(\beta \right)\left(-\tan\left(\beta \right)\right)\left(-\cos\left(\beta \right)\right)}{\cos\left(-\beta \right)\sin\left(360^{\circ}+\beta \right)}\) - step4: Transform the expression: \(\frac{-\sin\left(\beta \right)\left(-\tan\left(\beta \right)\right)\left(-\cos\left(\beta \right)\right)}{\cos\left(\beta \right)\sin\left(360^{\circ}+\beta \right)}\) - step5: Rewrite the expression: \(\frac{-\sin\left(\beta \right)\left(-\tan\left(\beta \right)\right)\left(-\cos\left(\beta \right)\right)}{\cos\left(\beta \right)\sin\left(\beta \right)}\) - step6: Reduce the fraction: \(-\frac{\sin\left(\beta \right)\left(-\tan\left(\beta \right)\right)\left(-\cos\left(\beta \right)\right)}{\cos\left(\beta \right)\sin\left(\beta \right)}\) - step7: Reduce the fraction: \(-\frac{-\tan\left(\beta \right)\left(-\cos\left(\beta \right)\right)}{\cos\left(\beta \right)}\) - step8: Factor: \(-\frac{-\tan\left(\beta \right)\left(-\cos\left(\beta \right)\right)}{\cos\left(\beta \right)\times 1}\) - step9: Factor the expression: \(-\frac{-\tan\left(\beta \right)\left(-1\right)\cos\left(\beta \right)}{\cos\left(\beta \right)}\) - step10: Reduce the fraction: \(-\left(-\tan\left(\beta \right)\left(-1\right)\right)\) - step11: Calculate: \(-\tan\left(\beta \right)\) Calculate or simplify the expression \( \frac { \cos ( \alpha - 90 ^ { \circ } ) \cdot \tan ( - \alpha ) } { \sin ( - \alpha ) \cdot \tan ( 720 ^ { \circ } - \alpha ) } \). Simplify the expression by following steps: - step0: Solution: \(\frac{\cos\left(\alpha -90^{\circ}\right)\tan\left(-\alpha \right)}{\sin\left(-\alpha \right)\tan\left(720^{\circ}-\alpha \right)}\) - step1: Rewrite the expression: \(\frac{\sin\left(\alpha \right)\tan\left(-\alpha \right)}{\sin\left(-\alpha \right)\tan\left(720^{\circ}-\alpha \right)}\) - step2: Transform the expression: \(\frac{\sin\left(\alpha \right)\left(-\tan\left(\alpha \right)\right)}{\sin\left(-\alpha \right)\tan\left(720^{\circ}-\alpha \right)}\) - step3: Transform the expression: \(\frac{\sin\left(\alpha \right)\left(-\tan\left(\alpha \right)\right)}{-\sin\left(\alpha \right)\tan\left(720^{\circ}-\alpha \right)}\) - step4: Rewrite the expression: \(\frac{\sin\left(\alpha \right)\left(-\tan\left(\alpha \right)\right)}{-\sin\left(\alpha \right)\left(-\tan\left(\alpha \right)\right)}\) - step5: Reduce the fraction: \(-\frac{\sin\left(\alpha \right)\left(-\tan\left(\alpha \right)\right)}{\sin\left(\alpha \right)\left(-\tan\left(\alpha \right)\right)}\) - step6: Reduce the fraction: \(-1\) Let's break down the two expressions step by step. ### Expression 5.2.5 The expression is: \[ \frac { \cos ( \alpha - 90 ^ { \circ } ) \cdot \tan ( - \alpha ) } { \sin ( - \alpha ) \cdot \tan ( 720 ^ { \circ } - \alpha ) } \] 1. **Simplifying \( \cos ( \alpha - 90^\circ ) \)**: \[ \cos ( \alpha - 90^\circ ) = \sin(\alpha) \] 2. **Simplifying \( \tan(-\alpha) \)**: \[ \tan(-\alpha) = -\tan(\alpha) \] 3. **Simplifying \( \sin(-\alpha) \)**: \[ \sin(-\alpha) = -\sin(\alpha) \] 4. **Simplifying \( \tan(720^\circ - \alpha) \)**: \[ \tan(720^\circ - \alpha) = \tan(-\alpha) = -\tan(\alpha) \] Now substituting these simplifications back into the expression: \[ \frac { \sin(\alpha) \cdot (-\tan(\alpha)) } { -\sin(\alpha) \cdot (-\tan(\alpha)) } \] This simplifies to: \[ \frac { \sin(\alpha) \cdot \tan(\alpha) } { \sin(\alpha) \cdot \tan(\alpha) } = -1 \] ### Expression 5.2.6 The expression is: \[ \frac { \sin ( \beta - 180 ^ { \circ } ) \cdot \tan ( - \beta - 180 ^ { \circ } ) \cdot \cos ( 180 ^ { \circ } + \beta ) } { \cos ( - \beta ) \cdot \sin ( 360 ^ { \circ } + \beta ) } \] 1. **Simplifying \( \sin ( \beta - 180^\circ ) \)**: \[ \sin ( \beta - 180^\circ ) = -\sin(\beta) \] 2. **Simplifying \( \tan(-\beta - 180^\circ) \)**: \[ \tan(-\beta - 180^\circ) = \tan(-\beta) = -\tan(\beta) \] 3. **Simplifying \( \cos(180^\circ + \beta) \)**: \[ \cos(180^\circ + \beta) = -\cos(\beta) \] 4. **Simplifying \( \cos(-\beta) \)**: \[ \cos(-\beta) = \cos(\beta) \] 5. **Simplifying \( \sin(360^\circ + \beta) \)**: \[ \sin(360^\circ + \beta) = \sin(\beta) \] Now substituting these simplifications back into the expression: \[ \frac { (-\sin(\beta)) \cdot (-\tan(\beta)) \cdot (-\cos(\beta)) } { \cos(\beta) \cdot \sin(\beta) } \] This simplifies to: \[ \frac { \sin(\beta) \cdot \tan(\beta) \cdot \cos(\beta) } { \cos(\beta) \cdot \sin(\beta) } = -\tan(\beta) \] ### Final Results - For Expression 5.2.5: The result is \(-1\). - For Expression 5.2.6: The result is \(-\tan(\beta)\).