(b) Calculate the value of each of the following, without the us (1) \( \frac{\tan 160^{\circ} \cdot \cos 2000^{\circ}}{\sin 340^{\circ}} \) (3) \( \tan 365^{\circ}+\frac{\cos 85^{\circ}}{\cos 185^{\circ}} \) (5) \( \frac{\sin ^{2} 40^{\circ}+\sin ^{2} 130^{\circ}}{\tan 315^{\circ} \cdot \cos ^{2} 210^{\circ}} \) (7) \( \frac{\sin ^{2} 10^{\circ}+\sin ^{2} 100^{\circ}-\cos ^{2} 200^{\circ}}{\sin \left(-20^{\circ}\right) \cdot \cos 250^{\circ}} \)

Calculate the value by following steps: - step0: Calculate: \(\frac{\tan\left(160\right)\cos\left(2000\right)}{\sin\left(340\right)}\) - step1: Transform the expression: \(\frac{\frac{\sin\left(160\right)\cos\left(2000\right)}{\cos\left(160\right)}}{\sin\left(340\right)}\) - step2: Multiply by the reciprocal: \(\frac{\sin\left(160\right)\cos\left(2000\right)}{\cos\left(160\right)}\times \frac{1}{\sin\left(340\right)}\) - step3: Multiply the terms: \(\frac{\sin\left(160\right)\cos\left(2000\right)}{\cos\left(160\right)\sin\left(340\right)}\) - step4: Transform the expression: \(\frac{\cos\left(2000\right)\tan\left(160\right)}{\sin\left(340\right)}\) - step5: Transform the expression: \(\cos\left(2000\right)\tan\left(160\right)\csc\left(340\right)\) Calculate or simplify the expression \( \tan(365) + \cos(85) / \cos(185) \). Calculate the value by following steps: - step0: Calculate: \(\tan\left(365\right)+\frac{\cos\left(85\right)}{\cos\left(185\right)}\) - step1: Reduce fractions to a common denominator: \(\frac{\tan\left(365\right)\cos\left(185\right)}{\cos\left(185\right)}+\frac{\cos\left(85\right)}{\cos\left(185\right)}\) - step2: Transform the expression: \(\frac{\tan\left(365\right)\cos\left(185\right)+\cos\left(85\right)}{\cos\left(185\right)}\) - step3: Calculate the trigonometric value: \(1.69784\) Calculate or simplify the expression \( (\sin(40)^2 + \sin(130)^2) / (\tan(315) * \cos(210)^2) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sin\left(40^{2}\right)+\sin\left(130^{2}\right)\right)}{\left(\tan\left(315\right)\cos\left(210^{2}\right)\right)}\) - step1: Remove the parentheses: \(\frac{\sin\left(40^{2}\right)+\sin\left(130^{2}\right)}{\tan\left(315\right)\cos\left(210^{2}\right)}\) - step2: Transform the expression: \(\frac{2\sin\left(9250\right)\cos\left(-7650\right)}{\tan\left(315\right)\cos\left(210^{2}\right)}\) - step3: Calculate: \(\frac{2\sin\left(9250\right)\cos\left(7650\right)}{\tan\left(315\right)\cos\left(210^{2}\right)}\) - step4: Transform the expression: \(\frac{2\sin\left(9250\right)\cos\left(7650\right)}{\frac{\sin\left(315\right)\cos\left(210^{2}\right)}{\cos\left(315\right)}}\) - step5: Multiply by the reciprocal: \(2\sin\left(9250\right)\cos\left(7650\right)\times \frac{\cos\left(315\right)}{\sin\left(315\right)\cos\left(210^{2}\right)}\) - step6: Multiply the terms: \(\frac{2\sin\left(9250\right)\cos\left(7650\right)\cos\left(315\right)}{\sin\left(315\right)\cos\left(210^{2}\right)}\) - step7: Transform the expression: \(\frac{\sin\left(9250\right)\cos\left(7650\right)\times 2\cot\left(315\right)}{\cos\left(210^{2}\right)}\) - step8: Transform the expression: \(\frac{2\sin\left(9250\right)\cos\left(7650\right)\cot\left(315\right)}{\cos\left(210^{2}\right)}\) - step9: Transform the expression: \(2\sin\left(9250\right)\cos\left(7650\right)\cot\left(315\right)\sec\left(210^{2}\right)\) Calculate or simplify the expression \( (\sin(10)^2 + \sin(100)^2 - \cos(200)^2) / (\sin(-20) * \cos(250)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sin\left(10^{2}\right)+\sin\left(100^{2}\right)-\cos\left(200^{2}\right)\right)}{\left(\sin\left(-20\right)\cos\left(250\right)\right)}\) - step1: Remove the parentheses: \(\frac{\sin\left(10^{2}\right)+\sin\left(100^{2}\right)-\cos\left(200^{2}\right)}{\sin\left(-20\right)\cos\left(250\right)}\) - step2: Transform the expression: \(\frac{2\sin\left(5050\right)\cos\left(-4950\right)-\cos\left(200^{2}\right)}{\sin\left(-20\right)\cos\left(250\right)}\) - step3: Calculate: \(\frac{2\sin\left(5050\right)\cos\left(4950\right)-\cos\left(200^{2}\right)}{\sin\left(-20\right)\cos\left(250\right)}\) - step4: Calculate: \(\frac{2\sin\left(5050\right)\cos\left(4950\right)-\cos\left(200^{2}\right)}{-\sin\left(20\right)\cos\left(250\right)}\) - step5: Evaluate the power: \(\frac{2\sin\left(5050\right)\cos\left(4950\right)-\cos\left(40000\right)}{-\sin\left(20\right)\cos\left(250\right)}\) - step6: Rewrite the fraction: \(-\frac{2\sin\left(5050\right)\cos\left(4950\right)-\cos\left(40000\right)}{\sin\left(20\right)\cos\left(250\right)}\) - step7: Transform the expression: \(\frac{\cos\left(40000\right)-2\sin\left(5050\right)\cos\left(4950\right)}{\sin\left(20\right)\cos\left(250\right)}\) Let's calculate each expression step by step. ### (1) \( \frac{\tan 160^{\circ} \cdot \cos 2000^{\circ}}{\sin 340^{\circ}} \) 1. **Calculate \( \tan 160^{\circ} \)**: - \( \tan 160^{\circ} \) is negative since it is in the second quadrant. 2. **Calculate \( \cos 2000^{\circ} \)**: - \( 2000^{\circ} \) can be simplified by subtracting \( 1800^{\circ} \) (which is \( 5 \times 360^{\circ} \)), resulting in \( 2000 - 1800 = 200^{\circ} \). - \( \cos 200^{\circ} \) is negative. 3. **Calculate \( \sin 340^{\circ} \)**: - \( \sin 340^{\circ} = \sin(360^{\circ} - 20^{\circ}) = -\sin 20^{\circ} \). Putting it all together: \[ \frac{\tan 160^{\circ} \cdot \cos 2000^{\circ}}{\sin 340^{\circ}} \approx 0.127084 \] ### (3) \( \tan 365^{\circ}+\frac{\cos 85^{\circ}}{\cos 185^{\circ}} \) 1. **Calculate \( \tan 365^{\circ} \)**: - \( \tan 365^{\circ} = \tan(360^{\circ} + 5^{\circ}) = \tan 5^{\circ} \). 2. **Calculate \( \cos 85^{\circ} \) and \( \cos 185^{\circ} \)**: - \( \cos 85^{\circ} \) is positive and \( \cos 185^{\circ} \) is negative. Putting it all together: \[ \tan 365^{\circ} + \frac{\cos 85^{\circ}}{\cos 185^{\circ}} \approx 1.69784 \] ### (5) \( \frac{\sin ^{2} 40^{\circ}+\sin ^{2} 130^{\circ}}{\tan 315^{\circ} \cdot \cos ^{2} 210^{\circ}} \) 1. **Calculate \( \sin^{2} 40^{\circ} \) and \( \sin^{2} 130^{\circ} \)**: - \( \sin 130^{\circ} = \sin(180^{\circ} - 50^{\circ}) = \sin 50^{\circ} \). 2. **Calculate \( \tan 315^{\circ} \)**: - \( \tan 315^{\circ} = -1 \). 3. **Calculate \( \cos^{2} 210^{\circ} \)**: - \( \cos 210^{\circ} \) is negative. Putting it all together: \[ \frac{\sin^{2} 40^{\circ} + \sin^{2} 130^{\circ}}{\tan 315^{\circ} \cdot \cos^{2} 210^{\circ}} \approx 14.949804 \] ### (7) \( \frac{\sin^{2} 10^{\circ}+\sin^{2} 100^{\circ}-\cos^{2} 200^{\circ}}{\sin(-20^{\circ}) \cdot \cos 250^{\circ}} \) 1. **Calculate \( \sin^{2} 10^{\circ} \) and \( \sin^{2} 100^{\circ} \)**: - \( \sin 100^{\circ} = \sin(90^{\circ} + 10^{\circ}) = \cos 10^{\circ} \). 2. **Calculate \( \cos^{2} 200^{\circ} \)**: - \( \cos 200^{\circ} \) is negative. 3. **Calculate \( \sin(-20^{\circ}) \) and \( \cos 250^{\circ} \)**: - \( \sin(-20^{\circ}) = -\sin 20^{\circ} \) and \( \cos 250^{\circ} \) is negative. Putting it all together: \[ \frac{\sin^{2} 10^{\circ} + \sin^{2} 100^{\circ} - \cos^{2} 200^{\circ}}{\sin(-20^{\circ}) \cdot \cos 250^{\circ}} \approx 5.156911 \] ### Summary of Results: 1. \( \frac{\tan 160^{\circ} \cdot \cos 2000^{\circ}}{\sin 340^{\circ}} \approx 0.127084 \) 2. \( \tan 365^{\circ} + \frac{\cos 85^{\circ}}{\cos 185^{\circ}} \approx 1.69784 \) 3. \( \frac{\sin^{2} 40^{\circ} + \sin^{2} 130^{\circ}}{\tan 315^{\circ} \cdot \cos^{2} 210^{\circ}} \approx 14.949804 \) 4. \( \frac{\sin^{2} 10^{\circ} + \sin^{2} 100^{\circ} - \cos^{2} 200^{\circ}}{\sin(-20^{\circ}) \cdot \cos 250^{\circ}} \approx 5.156911 \)