4. Simplify the following expressions without using a calculator. a \( \frac{\sin \left(-160^{\circ}\right) \sin \left(x-180^{\circ}\right)}{\cos \left(360^{\circ}-x\right) \tan (360+x) \cos 110^{\circ}} \) b \( \sin \left(90^{\circ}-x\right) \cos \left(180^{\circ}-x\right)+\tan \left(180^{\circ}+x\right) \cos (-x) \sin \left(360^{\circ}-x\right) \) c \( \frac{\cos \left(90^{\circ}+x\right) \tan 150^{\circ}}{\sin \left(180^{\circ}-x\right) \cos \left(-330^{\circ}\right)} \) d \( \frac{\cos \left(-210^{\circ}\right) \cos 120^{\circ}}{\sin 120^{\circ}}+\frac{\sin 135^{\circ} \tan ^{2} 240^{\circ}}{\cos 405^{\circ}} \)

Simplify the expression by following steps: - step0: Solution: \(\sin\left(90^{\circ}-x\right)\cos\left(180^{\circ}-x\right)+\tan\left(180^{\circ}+x\right)\cos\left(-x\right)\sin\left(360^{\circ}-x\right)\) - step1: Rewrite the expression: \(\cos\left(x\right)\cos\left(180^{\circ}-x\right)+\tan\left(180^{\circ}+x\right)\cos\left(-x\right)\sin\left(360^{\circ}-x\right)\) - step2: Rewrite the expression: \(\cos\left(x\right)\left(-\cos\left(x\right)\right)+\tan\left(180^{\circ}+x\right)\cos\left(-x\right)\sin\left(360^{\circ}-x\right)\) - step3: Rewrite the expression: \(\cos\left(x\right)\left(-\cos\left(x\right)\right)+\tan\left(x\right)\cos\left(-x\right)\sin\left(360^{\circ}-x\right)\) - step4: Transform the expression: \(\cos\left(x\right)\left(-\cos\left(x\right)\right)+\tan\left(x\right)\cos\left(x\right)\sin\left(360^{\circ}-x\right)\) - step5: Rewrite the expression: \(\cos\left(x\right)\left(-\cos\left(x\right)\right)+\tan\left(x\right)\cos\left(x\right)\left(-\sin\left(x\right)\right)\) - step6: Multiply the terms: \(-\cos^{2}\left(x\right)+\tan\left(x\right)\cos\left(x\right)\left(-\sin\left(x\right)\right)\) - step7: Multiply: \(-\cos^{2}\left(x\right)-\tan\left(x\right)\cos\left(x\right)\sin\left(x\right)\) - step8: Transform the expression: \(-\cos\left(x\right)\left(\cos\left(x\right)+\tan\left(x\right)\sin\left(x\right)\right)\) - step9: Transform the expression: \(-\cos\left(x\right)\times \frac{\cos^{2}\left(x\right)+\sin^{2}\left(x\right)}{\cos\left(x\right)}\) - step10: Reduce the fraction: \(-1\times \left(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\right)\) - step11: Multiply the terms: \(-\left(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\right)\) - step12: Calculate: \(-\cos^{2}\left(x\right)-\sin^{2}\left(x\right)\) - step13: Transform the expression: \(-\cos^{2}\left(x\right)-\left(1-\cos^{2}\left(x\right)\right)\) - step14: Calculate: \(-\cos^{2}\left(x\right)-1+\cos^{2}\left(x\right)\) - step15: Collect like terms: \(-1\) Calculate or simplify the expression \( \frac{\cos \left(-210^{\circ}\right) \cos 120^{\circ}}{\sin 120^{\circ}}+\frac{\sin 135^{\circ} \tan ^{2} 240^{\circ}}{\cos 405^{\circ}} \). Calculate the value by following steps: - step0: Calculate: \(\frac{\cos\left(-210^{\circ}\right)\cos\left(120^{\circ}\right)}{\sin\left(120^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(240^{\circ}\right)}{\cos\left(405^{\circ}\right)}\) - step1: Calculate: \(\frac{\cos\left(-210^{\circ}\right)\cos\left(120^{\circ}\right)}{\sin\left(120^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)}{\cos\left(405^{\circ}\right)}\) - step2: Calculate: \(\frac{\cos\left(210^{\circ}\right)\cos\left(120^{\circ}\right)}{\sin\left(120^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)}{\cos\left(405^{\circ}\right)}\) - step3: Multiply the terms: \(\frac{\frac{\sqrt{3}}{4}}{\sin\left(120^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)}{\cos\left(405^{\circ}\right)}\) - step4: Calculate: \(\frac{\frac{\sqrt{3}}{4}}{\sin\left(120^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)}{\cos\left(45^{\circ}\right)}\) - step5: Divide the terms: \(\frac{1}{2}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)}{\cos\left(45^{\circ}\right)}\) - step6: Reduce fractions to a common denominator: \(\frac{\cos\left(45^{\circ}\right)}{2\cos\left(45^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)\times 2}{\cos\left(45^{\circ}\right)\times 2}\) - step7: Reorder the terms: \(\frac{\cos\left(45^{\circ}\right)}{2\cos\left(45^{\circ}\right)}+\frac{\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)\times 2}{2\cos\left(45^{\circ}\right)}\) - step8: Transform the expression: \(\frac{\cos\left(45^{\circ}\right)+\sin\left(135^{\circ}\right)\tan^{2}\left(60^{\circ}\right)\times 2}{2\cos\left(45^{\circ}\right)}\) - step9: Multiply the terms: \(\frac{\cos\left(45^{\circ}\right)+\sqrt{2}\times \tan^{2}\left(60^{\circ}\right)}{2\cos\left(45^{\circ}\right)}\) - step10: Calculate the trigonometric value: \(\frac{\frac{\sqrt{2}}{2}+\sqrt{2}\times \tan^{2}\left(60^{\circ}\right)}{2\cos\left(45^{\circ}\right)}\) - step11: Rewrite the expression: \(\frac{\frac{\sqrt{2}+2\sqrt{2}\times \tan^{2}\left(60^{\circ}\right)}{2}}{2\cos\left(45^{\circ}\right)}\) - step12: Multiply by the reciprocal: \(\frac{\sqrt{2}+2\sqrt{2}\times \tan^{2}\left(60^{\circ}\right)}{2}\times \frac{1}{2\cos\left(45^{\circ}\right)}\) - step13: Multiply the terms: \(\frac{\sqrt{2}+2\sqrt{2}\times \tan^{2}\left(60^{\circ}\right)}{2\sqrt{2}}\) - step14: Factor: \(\frac{\sqrt{2}\times \left(1+2\tan^{2}\left(60^{\circ}\right)\right)}{2\sqrt{2}}\) - step15: Reduce the fraction: \(\frac{1+2\tan^{2}\left(60^{\circ}\right)}{2}\) - step16: Transform the expression: \(\frac{7}{2}\) Calculate or simplify the expression \( \frac{\cos \left(90^{\circ}+x\right) \tan 150^{\circ}}{\sin \left(180^{\circ}-x\right) \cos \left(-330^{\circ}\right)} \). Simplify the expression by following steps: - step0: Solution: \(\frac{\cos\left(90^{\circ}+x\right)\tan\left(150^{\circ}\right)}{\sin\left(180^{\circ}-x\right)\cos\left(-330^{\circ}\right)}\) - step1: Rewrite the expression: \(\frac{-\sin\left(x\right)\tan\left(150^{\circ}\right)}{\sin\left(180^{\circ}-x\right)\cos\left(-330^{\circ}\right)}\) - step2: Rewrite the expression: \(\frac{-\sin\left(x\right)\tan\left(150^{\circ}\right)}{\sin\left(x\right)\cos\left(-330^{\circ}\right)}\) - step3: Calculate: \(\frac{-\sin\left(x\right)\tan\left(150^{\circ}\right)}{\sin\left(x\right)\cos\left(330^{\circ}\right)}\) - step4: Multiply the terms: \(\frac{-\sin\left(x\right)\tan\left(150^{\circ}\right)}{\frac{\sqrt{3}}{2}\sin\left(x\right)}\) - step5: Reduce the fraction: \(-\frac{\sin\left(x\right)\tan\left(150^{\circ}\right)}{\frac{\sqrt{3}}{2}\sin\left(x\right)}\) - step6: Reduce the fraction: \(-\frac{\tan\left(150^{\circ}\right)}{\frac{\sqrt{3}}{2}}\) - step7: Divide the terms: \(-\left(-\frac{2\sqrt{3}}{3\sqrt{3}}\right)\) - step8: Calculate: \(\frac{2\sqrt{3}}{3\sqrt{3}}\) - step9: Reduce the fraction: \(\frac{2}{3}\) Calculate or simplify the expression \( \frac{\sin \left(-160^{\circ}\right) \sin \left(x-180^{\circ}\right)}{\cos \left(360^{\circ}-x\right) \tan (360+x) \cos 110^{\circ}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{\sin\left(-160^{\circ}\right)\sin\left(x-180^{\circ}\right)}{\cos\left(360^{\circ}-x\right)\tan\left(360+x\right)\cos\left(110^{\circ}\right)}\) - step1: Calculate: \(\frac{-\sin\left(160^{\circ}\right)\sin\left(x-180^{\circ}\right)}{\cos\left(360^{\circ}-x\right)\tan\left(360+x\right)\cos\left(110^{\circ}\right)}\) - step2: Rewrite the expression: \(\frac{-\sin\left(160^{\circ}\right)\left(-\sin\left(x\right)\right)}{\cos\left(360^{\circ}-x\right)\tan\left(360+x\right)\cos\left(110^{\circ}\right)}\) - step3: Rewrite the expression: \(\frac{-\sin\left(160^{\circ}\right)\left(-\sin\left(x\right)\right)}{\cos\left(x\right)\tan\left(360+x\right)\cos\left(110^{\circ}\right)}\) - step4: Rewrite the expression: \(\frac{-\sin\left(160^{\circ}\right)\left(-\sin\left(x\right)\right)}{\cos\left(x\right)\tan\left(x+360\right)\cos\left(110^{\circ}\right)}\) - step5: Rewrite the expression: \(\frac{\sin\left(160^{\circ}\right)\sin\left(x\right)}{\cos\left(x\right)\tan\left(x+360\right)\cos\left(110^{\circ}\right)}\) - step6: Multiply the terms: \(\frac{\sin\left(160^{\circ}\right)\sin\left(x\right)}{-\cos\left(70^{\circ}\right)\cos\left(x\right)\tan\left(x+360\right)}\) - step7: Reduce the fraction: \(-\frac{\sin\left(160^{\circ}\right)\sin\left(x\right)}{\cos\left(70^{\circ}\right)\cos\left(x\right)\tan\left(x+360\right)}\) - step8: Reduce the fraction: \(-\frac{\sin\left(x\right)}{\cos\left(x\right)\tan\left(x+360\right)}\) - step9: Transform the expression: \(-\frac{\sin\left(x\right)\left(1-\tan\left(360\right)\tan\left(x\right)\right)}{\cos\left(x\right)\left(\tan\left(360\right)+\tan\left(x\right)\right)}\) - step10: Transform the expression: \(-\frac{\sin\left(x\right)\left(\cos\left(360\right)\cos\left(x\right)-\sin\left(360\right)\sin\left(x\right)\right)}{\cos\left(x\right)\left(\sin\left(360\right)\cos\left(x\right)+\cos\left(360\right)\sin\left(x\right)\right)}\) - step11: Rewrite the expression: \(-\frac{\sin\left(x\right)\cos\left(x+360\right)}{\cos\left(x\right)\sin\left(x+360\right)}\) - step12: Transform the expression: \(-\tan\left(x\right)\cot\left(x+360\right)\) Let's simplify each expression step by step. ### a) \[ \frac{\sin \left(-160^{\circ}\right) \sin \left(x-180^{\circ}\right)}{\cos \left(360^{\circ}-x\right) \tan (360+x) \cos 110^{\circ}} \] 1. **Simplify \(\sin(-160^{\circ})\)**: \[ \sin(-\theta) = -\sin(\theta) \implies \sin(-160^{\circ}) = -\sin(160^{\circ}) \] 2. **Simplify \(\sin(x-180^{\circ})\)**: \[ \sin(x-180^{\circ}) = -\sin(x) \] 3. **Simplify \(\cos(360^{\circ}-x)\)**: \[ \cos(360^{\circ}-x) = \cos(x) \] 4. **Simplify \(\tan(360+x)\)**: \[ \tan(360+x) = \tan(x) \] 5. **Combine everything**: \[ \frac{-\sin(160^{\circ})(-\sin(x))}{\cos(x) \tan(x) \cos(110^{\circ})} = \frac{\sin(160^{\circ} \sin(x)}{\cos(x) \tan(x) \cos(110^{\circ})} \] 6. **Final result**: \[ -\tan(x) \cot(x+360) \] ### b) \[ \sin \left(90^{\circ}-x\right) \cos \left(180^{\circ}-x\right)+\tan \left(180^{\circ}+x\right) \cos (-x) \sin \left(360^{\circ}-x\right) \] 1. **Simplify \(\sin(90^{\circ}-x)\)**: \[ \sin(90^{\circ}-x) = \cos(x) \] 2. **Simplify \(\cos(180^{\circ}-x)\)**: \[ \cos(180^{\circ}-x) = -\cos(x) \] 3. **Simplify \(\tan(180^{\circ}+x)\)**: \[ \tan(180^{\circ}+x) = \tan(x) \] 4. **Simplify \(\cos(-x)\)**: \[ \cos(-x) = \cos(x) \] 5. **Simplify \(\sin(360^{\circ}-x)\)**: \[ \sin(360^{\circ}-x) = -\sin(x) \] 6. **Combine everything**: \[ \cos(x)(-\cos(x)) + \tan(x)(\cos(x))(-\sin(x)) = -\cos^2(x) - \tan(x)\cos(x)\sin(x) \] 7. **Final result**: \[ -1 \] ### c) \[ \frac{\cos \left(90^{\circ}+x\right) \tan 150^{\circ}}{\sin \left(180^{\circ}-x\right) \cos \left(-330^{\circ}\right)} \] 1. **Simplify \(\cos(90^{\circ}+x)\)**: \[ \cos(90^{\circ}+x) = -\sin(x) \] 2. **Simplify \(\tan(150^{\circ})\)**: \[ \tan(150^{\circ}) = -\tan(30^{\circ}) = -\frac{1}{\sqrt{3}} \] 3. **Simplify \(\sin(180^{\circ}-x)\)**: \[ \sin(180^{\circ}-x) = \sin(x) \] 4. **Simplify \(\cos(-330^{\circ})\)**: \[ \cos(-330^{\circ}) = \cos(330^{\circ}) = \frac{\sqrt{3}}{2} \] 5. **Combine everything**: \[ \frac{-\sin(x)(-\frac{1}{\sqrt{3}})}{\sin(x)(\frac{\sqrt{3}}{2})} = \frac{\frac{\sin(x)}{\sqrt{3}}}{\sin(x)\frac{\sqrt{3}}{2}} = \frac{2}{3} \] ### d) \[ \frac{\cos \left(-210^{\circ}\right) \cos 120^{\circ}}{\sin 120^{\circ}}+\frac{\sin 135^{\circ} \tan ^{2} 240^{\circ}}{\cos 405^{\circ}} \] 1. **Simplify \(\cos(-210^{\circ})\)**: \[ \cos(-210^{\circ}) = \cos(210^{\circ}) = -\frac{\sqrt{3}}{2} \] 2. **Simplify \(\cos(120^{\circ})\)**: \[ \cos(120^{\circ}) = -\frac{1}{2} \] 3. **Simplify \(\sin(120^{\circ})\)**: \[ \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \] 4. **Simplify \(\sin(135^{\circ})\)**: \[ \sin(135^{\circ}) = \frac{\sqrt{2}}{2} \] 5. **Simplify \(\tan(240^{\circ})\)**: \[ \tan(240^{\circ}) = \tan(60^{\circ}) = \sqrt{3} \implies \tan^2(240^{\circ}) = 3 \] 6. **Simplify \(\cos(405^{\circ})\)**: \[ \cos(405^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \] 7. **Combine everything**: \[ \frac{-\frac{\sqrt{3}}{2} \cdot -\frac{1}{2}}{\frac{\sqrt{3}}{2}} + \frac{\frac{\sqrt{2}}{2} \cdot 3}{\frac{\sqrt{2}}{2}} = \frac{\frac{\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}} + 3 = \frac{1}{2} + 3 = \frac{7}{2} \] ### Summary of Results: - a