3. Вычислите: a) 2sin \(\frac{5\pi}{12}\)cos \(\frac{5\pi}{12}\); 6) cos² \(\frac{\pi}{12}\) - sin² \(\frac{\pi}{12}\); в) 2cos²15°tg15°; г) (cos22,5° - sin22,5°)².

a) Используем формулу синуса двойного угла: 2sin α cos α = sin 2α. Тогда:

2sin \(\frac{5\pi}{12}\)cos \(\frac{5\pi}{12}\) = sin(2 \(\cdot\) \(\frac{5\pi}{12}\)) = sin(\(\frac{5\pi}{6}\)) = sin(150°) = \(\frac{1}{2}\).

Ответ: \(\frac{1}{2}\)

б) Используем формулу косинуса двойного угла: cos² α - sin² α = cos 2α. Тогда:

cos² \(\frac{\pi}{12}\) - sin² \(\frac{\pi}{12}\) = cos(2 \(\cdot\) \(\frac{\pi}{12}\)) = cos(\(\frac{\pi}{6}\)) = cos(30°) = \(\frac{\sqrt{3}}{2}\).

Ответ: \(\frac{\sqrt{3}}{2}\)

в) Используем формулу: 2cos² α = 1 + cos 2α.

tg15° = \(\frac{sin15°}{cos15°}\).

2cos²15°tg15° = (1 + cos30°) \(\frac{sin15°}{cos15°}\) = (1 + \(\frac{\sqrt{3}}{2}\)) \(\frac{sin15°}{cos15°}\) = \(\frac{(2 + \sqrt{3})sin15°}{2cos15°}\)

Используем формулы sin(α - β) = sin α cos β - cos α sin β и cos(α - β) = cos α cos β + sin α sin β.

sin15° = sin(45° - 30°) = sin45°cos30° - cos45°sin30° = \(\frac{\sqrt{2}}{2}\) \(\cdot\) \(\frac{\sqrt{3}}{2}\) - \(\frac{\sqrt{2}}{2}\) \(\cdot\) \(\frac{1}{2}\) = \(\frac{\sqrt{6} - \sqrt{2}}{4}\).

cos15° = cos(45° - 30°) = cos45°cos30° + sin45°sin30° = \(\frac{\sqrt{2}}{2}\) \(\cdot\) \(\frac{\sqrt{3}}{2}\) + \(\frac{\sqrt{2}}{2}\) \(\cdot\) \(\frac{1}{2}\) = \(\frac{\sqrt{6} + \sqrt{2}}{4}\).

2cos²15°tg15° = \(\frac{(2 + \sqrt{3})(\sqrt{6} - \sqrt{2})}{2(\sqrt{6} + \sqrt{2})}\) = \(\frac{(2 + \sqrt{3})(\sqrt{3} - 1)}{2(\sqrt{3} + 1)}\) = \(\frac{(2 + \sqrt{3})(\sqrt{3} - 1)}{2(\sqrt{3} + 1)} \cdot \frac{(\sqrt{3} - 1)}{(\sqrt{3} - 1)}\) = \(\frac{(2 + \sqrt{3})(4 - 2\sqrt{3})}{2(2)}\) = (2 + \sqrt{3})(1 - \(\frac{\sqrt{3}}{2}\)) = 2 - \sqrt{3} + \sqrt{3} - \(\frac{3}{2}\) = \(\frac{1}{2}\).

Ответ: \(\frac{1}{2}\)

г) Используем формулу косинуса двойного угла: cos 2α = cos² α - sin² α.

(cos22,5° - sin22,5°)² = (cos(2 \(\cdot\) 22,5°))² = (cos45°)² = (\(\frac{\sqrt{2}}{2}\))² = \(\frac{1}{2}\).

Ответ: \(\frac{1}{2}\)