5.3 Simplify the following to a single trigonometric function, without the calculator. \( \frac{\cos \left(90^{\circ}-\alpha\right) \sin \left(-\alpha-540^{\circ}\right)}{\tan 225^{\circ}+\sin \alpha \cdot \sin \left(180^{\circ}+\alpha\right)} \)

To simplify the given expression, let's break it down step-by-step. 1. **Evaluate the components:** - \(\cos(90^{\circ}-\alpha) = \sin(\alpha)\) (using the co-function identity) - \(\sin(-\alpha - 540^{\circ})\): First, simplify \(-540^{\circ}\) to \(-540 + 720 = 180^{\circ}\). So, \(\sin(-\alpha - 540^{\circ}) = \sin(-\alpha + 180^{\circ}) = -\sin(\alpha)\) (since \(\sin(180^{\circ} + x) = -\sin(x)\)). - \(\tan(225^{\circ}) = \tan(180 + 45^{\circ}) = \tan(45^{\circ}) = -1\). - \(\sin(180^{\circ} + \alpha) = -\sin(\alpha)\). 2. **Substituting back into the expression:** The expression now looks like this: \[ \frac{\sin(\alpha)(-\sin(\alpha))}{-1 + \sin(\alpha)(-\sin(\alpha))} \] This simplifies to: \[ \frac{-\sin^2(\alpha)}{-1 - \sin^2(\alpha)} \] 3. **Further simplify the denominator:** The denominator \(-1 - \sin^2(\alpha) = - (1 + \sin^2(\alpha))\). Thus, the expression simplifies to: \[ \frac{-\sin^2(\alpha)}{- (1 + \sin^2(\alpha))} = \frac{\sin^2(\alpha)}{1 + \sin^2(\alpha)} \] 4. **Final Result:** So, the final simplified expression is: \[ \frac{\sin^2(\alpha)}{1 + \sin^2(\alpha)} \]