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1. Simplify: \( 1.1 \frac{\sin 40^{\circ} \cos 40^{\circ}}{\cos 120^{\circ} \cos 190^{\circ}} \) \( 1.2 \frac{\sin 54^{\circ}}{\sin 18^{\circ}}-\frac{\cos 54^{\circ}}{\cos 18^{\circ}} \) \( 1.3 \frac{\sin 33^{\circ}}{\sin 11^{\circ}}-\frac{\cos 33^{\circ}}{\cos 11^{\circ}} \) \( 1.41-4 \sin ^{2} 15^{\circ} \)

Ask by Harris Elliott. in South Africa
Mar 10,2025

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1.1. \( 1 \) 1.2. \( 2 \) 1.3. \( 2 \) 1.4. \( \sqrt{3} - 1 \)

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To simplify the expressions, we can apply trigonometric identities and values for the specific angles. First Line: For \( 1.1 \), use the fact that \(\cos 120^{\circ} = -\frac{1}{2}\) and \(\cos 190^{\circ} = -\cos 10^{\circ}\). So, the denominator becomes \(\frac{1}{4} \cos 10^{\circ}\), and using \( \sin 2\theta = 2\sin \theta \cos \theta \), \( 2\sin 40^{\circ} \cos 40^{\circ} = \sin 80^{\circ}\), and you can simplify to get the final value. Second Line: For \( 1.2 \), recognize that \(\sin 54^{\circ} = \cos 36^{\circ}\) and \(\cos 54^{\circ} = \sin 36^{\circ}\); substituting gives \(\frac{\cos 36^{\circ}}{\sin 18^{\circ}} - \frac{\sin 36^{\circ}}{\cos 18^{\circ}}\). This can be rearranged, and using the angle addition formulas could yield an even simpler result, perhaps leading to another trigonometric core identity.

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