1. \( \frac{\cos \left(180^{\circ}-x\right) \sin \left(x-90^{\circ}\right)-1}{\tan ^{2}\left(540^{\circ}+x\right) \sin \left(90^{\circ}+x\right) \cos (-x)} \) 2. \( \left[\sin (-\theta)+\cos \left(360^{\circ}+\theta\right)\right]\left[\cos \left(\theta-90^{\circ}\right)+\cos \left(180^{\circ}+\theta\right.\right. \) 3. \( \cos ^{2} \theta\left(1+\tan ^{2} \theta\right) \) 4. \( \frac{1-\cos ^{2} \theta}{1-\sin ^{2} \theta} \)

Simplify the expression by following steps: - step0: Solution: \(\cos\left(\theta ^{2}\right)\left(1+\tan\left(\theta ^{2}\right)\right)\) - step1: Transform the expression: \(\cos\left(\theta ^{2}\right)\times \frac{\cos\left(\theta ^{2}\right)+\sin\left(\theta ^{2}\right)}{\cos\left(\theta ^{2}\right)}\) - step2: Reduce the fraction: \(1\times \left(\cos\left(\theta ^{2}\right)+\sin\left(\theta ^{2}\right)\right)\) - step3: Multiply the terms: \(\cos\left(\theta ^{2}\right)+\sin\left(\theta ^{2}\right)\) Calculate or simplify the expression \( (1-\cos(\theta)^2)/(1-\sin(\theta)^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(1-\cos\left(\theta ^{2}\right)\right)}{\left(1-\sin\left(\theta ^{2}\right)\right)}\) - step1: Remove the parentheses: \(\frac{1-\cos\left(\theta ^{2}\right)}{1-\sin\left(\theta ^{2}\right)}\) Calculate or simplify the expression \( (\sin(-\theta)+\cos(360+\theta))*(\cos(\theta-90)+\cos(180+\theta)) \). Simplify the expression by following steps: - step0: Solution: \(\left(\sin\left(-\theta \right)+\cos\left(360+\theta \right)\right)\left(\cos\left(\theta -90\right)+\cos\left(180+\theta \right)\right)\) - step1: Transform the expression: \(\left(-\sin\left(\theta \right)+\cos\left(360+\theta \right)\right)\left(\cos\left(\theta -90\right)+\cos\left(180+\theta \right)\right)\) - step2: Rewrite the expression: \(\left(-\sin\left(\theta \right)+\cos\left(\theta +360\right)\right)\left(\cos\left(\theta -90\right)+\cos\left(180+\theta \right)\right)\) - step3: Transform the expression: \(\left(-\sin\left(\theta \right)+\cos\left(\theta +360\right)\right)\left(2\cos\left(\theta +45\right)\cos\left(-135\right)\right)\) - step4: Calculate: \(\left(-\sin\left(\theta \right)+\cos\left(\theta +360\right)\right)\left(2\cos\left(\theta +45\right)\cos\left(135\right)\right)\) - step5: Multiply the terms: \(\left(-\sin\left(\theta \right)+\cos\left(\theta +360\right)\right)\times 2\cos\left(135\right)\cos\left(\theta +45\right)\) - step6: Simplify: \(2\cos\left(135\right)\left(-\sin\left(\theta \right)+\cos\left(\theta +360\right)\right)\cos\left(\theta +45\right)\) - step7: Simplify: \(2\cos\left(135\right)\cos\left(\theta +45\right)\left(-\sin\left(\theta \right)+\cos\left(\theta +360\right)\right)\) - step8: Multiply the terms: \(\left(-2\sin\left(\theta \right)+2\cos\left(\theta +360\right)\right)\cos\left(135\right)\cos\left(\theta +45\right)\) - step9: Multiply the terms: \(\left(-2\cos\left(135\right)\sin\left(\theta \right)+2\cos\left(135\right)\cos\left(\theta +360\right)\right)\cos\left(\theta +45\right)\) Let's break down the problems step by step and summarize the results. ### 1. Expression: \[ \frac{\cos \left(180^{\circ}-x\right) \sin \left(x-90^{\circ}\right)-1}{\tan ^{2}\left(540^{\circ}+x\right) \sin \left(90^{\circ}+x\right) \cos (-x)} \] **Result:** The calculation timed out, indicating that it may be too complex to simplify directly. We can analyze it further if needed. ### 2. Expression: \[ \left[\sin (-\theta)+\cos \left(360^{\circ}+\theta\right)\right]\left[\cos \left(\theta-90^{\circ}\right)+\cos \left(180^{\circ}+\theta\right)\right] \] **Simplified Result:** \[ (-2\cos(135)\sin(\theta)+2\cos(135)\cos(\theta +360))\cos(\theta +45) \] ### 3. Expression: \[ \cos ^{2} \theta\left(1+\tan ^{2} \theta\right) \] **Simplified Result:** \[ \cos(\theta ^{2})+\sin(\theta ^{2}) \] ### 4. Expression: \[ \frac{1-\cos ^{2} \theta}{1-\sin ^{2} \theta} \] **Simplified Result:** \[ \frac{1-\cos(\theta ^{2})}{1-\sin(\theta ^{2})} \] ### Summary of Results: 1. The first expression requires further analysis due to timeout. 2. The second expression simplifies to \((-2\cos(135)\sin(\theta)+2\cos(135)\cos(\theta +360))\cos(\theta +45)\). 3. The third expression simplifies to \(\cos(\theta ^{2})+\sin(\theta ^{2})\). 4. The fourth expression simplifies to \(\frac{1-\cos(\theta ^{2})}{1-\sin(\theta ^{2})}\). If you need further assistance with the first expression or any other part, please let me know!