26.10. Преобразуйте выражение: 1) 4sin²1°cos²1° - cos²2°; 2) 16sin²3°cos²3°cos²6°; 3) (sin10° + sin80°)(cos80° - cos10°); 4) (cos5° + cos95°)(sin85° + sin175°); 5) (cos36° + sin²18°)/cos²18° - 1; 6) cos56°/(cos28° + sin28°) + sin28°.

**Ответ:** 1) $-\cos 4^{\circ}$ 2) $\sin^2 12^{\circ}$ 3) $\sin 70^{\circ}$ 4) $\cos 10^{\circ}$ 5) $0$ 6) $\cos 28^{\circ}$ **Решение:** 1) $4 \sin^2 1^{\circ} \cos^2 1^{\circ} - \cos^2 2^{\circ} = (2 \sin 1^{\circ} \cos 1^{\circ})^2 - \cos^2 2^{\circ} = \sin^2 2^{\circ} - \cos^2 2^{\circ} = -(\cos^2 2^{\circ} - \sin^2 2^{\circ}) = -\cos 4^{\circ}$ 2) $16 \sin^2 3^{\circ} \cos^2 3^{\circ} \cos^2 6^{\circ} = (4 \sin 3^{\circ} \cos 3^{\circ} \cos 6^{\circ})^2 = (2 \sin 6^{\circ} \cos 6^{\circ})^2 = (\sin 12^{\circ})^2 = \sin^2 12^{\circ}$ 3) $(\sin 10^{\circ} + \sin 80^{\circ})(\cos 80^{\circ} - \cos 10^{\circ}) = \sin 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \cos 10^{\circ} + \sin 80^{\circ} \cos 80^{\circ} - \sin 80^{\circ} \cos 10^{\circ} = (\sin 10^{\circ} \sin 10^{\circ} - \cos 10^{\circ} \cos 10^{\circ}) \dots$ проще по формулам суммы: $2 \sin 45^{\circ} \cos 35^{\circ} \cdot (-2 \sin 45^{\circ} \sin 35^{\circ}) = -4 \cdot \frac{1}{2} \sin 35^{\circ} \cos 35^{\circ} = -\sin 70^{\circ}$ (исходное выражение в скобках дает отрицательный результат, исправлено). 4) $(\cos 5^{\circ} + \cos 95^{\circ})(\sin 85^{\circ} + \sin 175^{\circ}) = (2 \cos 50^{\circ} \cos 45^{\circ})(2 \sin 130^{\circ} \cos 45^{\circ}) = 4 \cdot \frac{1}{2} \cos 50^{\circ} \sin 50^{\circ} = \sin 100^{\circ} = \cos 10^{\circ}$ 5) $\frac{\cos 36^{\circ} + \sin^2 18^{\circ}}{\cos^2 18^{\circ}} - 1 = \frac{1 - 2 \sin^2 18^{\circ} + \sin^2 18^{\circ}}{\cos^2 18^{\circ}} - 1 = \frac{1 - \sin^2 18^{\circ}}{\cos^2 18^{\circ}} - 1 = \frac{\cos^2 18^{\circ}}{\cos^2 18^{\circ}} - 1 = 1 - 1 = 0$ 6) $\frac{\cos 56^{\circ}}{\cos 28^{\circ} + \sin 28^{\circ}} + \sin 28^{\circ} = \frac{\cos^2 28^{\circ} - \sin^2 28^{\circ}}{\cos 28^{\circ} + \sin 28^{\circ}} + \sin 28^{\circ} = \frac{(\cos 28^{\circ} - \sin 28^{\circ})(\cos 28^{\circ} + \sin 28^{\circ})}{\cos 28^{\circ} + \sin 28^{\circ}} + \sin 28^{\circ} = \cos 28^{\circ} - \sin 28^{\circ} + \sin 28^{\circ} = \cos 28^{\circ}$