№3.1 Упростите выражения a) tg (0+2). Sin (80+2) 6)-td-sip cos cos'd 6) og + Sti²2+cos2 2) 0082-8712 +249d tga (1-8) (1+82)

a) tg (\(\alpha\)+\(\beta\)) \(\cdot\) sin(\(\frac{\pi}{2}\) + \(\alpha\)) / cos (\(\frac{\pi}{2}\) - \(\alpha\)) Используем формулы приведения: sin(\(\frac{\pi}{2}\) + \(\alpha\)) = cos(\(\alpha\)), cos (\(\frac{\pi}{2}\) - \(\alpha\)) = sin(\(\alpha\)) tg (\(\alpha\)+\(\beta\)) \(\cdot\) cos(\(\alpha\)) / sin(\(\alpha\)) = tg (\(\alpha\)+\(\beta\)) \(\cdot\) ctg(\(\alpha\)) = \(\frac{sin(\alpha+\beta)}{cos(\alpha+\beta)}\) \(\cdot\) \(\frac{cos(\alpha)}{sin(\alpha)}\) = \(\frac{sin(\alpha+\beta)cos(\alpha)}{cos(\alpha+\beta)sin(\alpha)}\)

б) \(\frac{1}{cos^2 \alpha}\) - tg²\(\alpha\) - sin²\(\beta\) = \(\frac{1}{cos^2 \alpha}\) - \(\frac{sin^2 \alpha}{cos^2 \alpha}\) - sin²\(\beta\) = \(\frac{1-sin^2 \alpha}{cos^2 \alpha}\) - sin²\(\beta\) = \(\frac{cos^2 \alpha}{cos^2 \alpha}\) - sin²\(\beta\) = 1 - sin²\(\beta\) = cos²\(\beta\)

в) ctg(\(\alpha\)) / tg(\(\alpha\)) + sin²(\(\alpha\)) + cos²(\(\alpha\)) = \(\frac{cos(\alpha)}{sin(\alpha)}\) / \(\frac{sin(\alpha)}{cos(\alpha)}\) + 1 = \(\frac{cos(\alpha)}{sin(\alpha)}\) \(\cdot\) \(\frac{cos(\alpha)}{sin(\alpha)}\) + 1 = \(\frac{cos^2(\alpha)}{sin^2(\alpha)}\) + 1 = ctg²(\(\alpha\)) + 1 = \(\frac{1}{sin^2(\alpha)}\)

г) \(\frac{cos^2(\alpha)-sin^2(\alpha)}{(1-sin(\alpha))(1+sin(\alpha))}\) + 2tg²(\(\alpha\) = \(\frac{cos^2(\alpha)-sin^2(\alpha)}{1-sin^2(\alpha)}\) + 2tg²(\(\alpha\) = \(\frac{cos^2(\alpha)-sin^2(\alpha)}{cos^2(\alpha)}\) + 2tg²(\(\alpha\) = \(\frac{cos^2(\alpha)}{cos^2(\alpha)}\) - \(\frac{sin^2(\alpha)}{cos^2(\alpha)}\) + 2tg²(\(\alpha\) = 1 - tg²(\(\alpha\)) + 2tg²(\(\alpha\) = 1 + tg²(\(\alpha\) = \(\frac{1}{cos^2(\alpha)}\)

Ответ: a) \(\frac{sin(\alpha+\beta)cos(\alpha)}{cos(\alpha+\beta)sin(\alpha)}\); б) cos²\(\beta\); в) \(\frac{1}{sin^2(\alpha)}\); г) \(\frac{1}{cos^2(\alpha)}\)