Можешь упростить тригонометрические выражения?

Конечно, давай упростим эти выражения! 1) $tg(\frac{\pi}{2} - \alpha) \cdot ctg(2\pi - \alpha) \cdot cos(\frac{\pi}{2} - \alpha) \cdot tg(2\pi + \alpha) = ctg(\alpha) \cdot ctg(-\alpha) \cdot sin(\alpha) \cdot tg(\alpha) = \frac{cos(\alpha)}{sin(\alpha)} \cdot (-\frac{cos(\alpha)}{sin(\alpha)}) \cdot sin(\alpha) \cdot \frac{sin(\alpha)}{cos(\alpha)} = -cos(\alpha)$. *Перевод: tg(π/2 - α) * ctg(2π - α) * cos(π/2 - α) * tg(2π + α) = ctg(α) * ctg(-α) * sin(α) * tg(α) = cos(α)/sin(α) * (-cos(α)/sin(α)) * sin(α) * sin(α)/cos(α) = -cos(α)* 2) $\frac{sin \alpha - sin 3 \alpha}{cos \alpha - cos 3 \alpha} = \frac{2cos(\frac{\alpha + 3\alpha}{2})sin(\frac{\alpha - 3\alpha}{2})}{ -2sin(\frac{\alpha + 3\alpha}{2})sin(\frac{\alpha - 3\alpha}{2})} = -ctg(2\alpha)$. *Перевод: (sin α - sin 3 α) / (cos α - cos 3 α) = 2cos((α + 3α)/2)sin((α - 3α)/2) / -2sin((α + 3α)/2)sin((α - 3α)/2) = -ctg(2α)* 3) $cos(2\pi - \alpha) \cdot (tg(\frac{3\pi}{2} - \alpha))^2 \cdot tg(2\pi + \alpha) \cdot (ctg(\frac{\pi}{2} + \alpha))^2 = cos(-\alpha) \cdot (ctg(\alpha))^2 \cdot tg(\alpha) \cdot (-tg(\alpha))^2 = cos(\alpha) \cdot \frac{cos^2(\alpha)}{sin^2(\alpha)} \cdot \frac{sin(\alpha)}{cos(\alpha)} \cdot tg^2(\alpha) = cos(\alpha) \cdot \frac{cos^2(\alpha)}{sin^2(\alpha)} \cdot \frac{sin(\alpha)}{cos(\alpha)} \cdot \frac{sin^2(\alpha)}{cos^2(\alpha)} = sin(\alpha)$. *Перевод: cos(2π - α) * (tg(3π/2 - α))^2 * tg(2π + α) * (ctg(π/2 + α))^2 = cos(-α) * (ctg(α))^2 * tg(α) * (-tg(α))^2 = cos(α) * (cos^2(α)/sin^2(α)) * (sin(α)/cos(α)) * tg^2(α) = cos(α) * (cos^2(α)/sin^2(α)) * (sin(α)/cos(α)) * (sin^2(α)/cos^2(α)) = sin(α)* 4) $\frac{sin^2 \alpha - tg^2 \alpha}{cos^2 \alpha - ctg^2 \alpha} = \frac{sin^2(\alpha) - \frac{sin^2(\alpha)}{cos^2(\alpha)}}{cos^2(\alpha) - \frac{cos^2(\alpha)}{sin^2(\alpha)}} = \frac{sin^2(\alpha)(1 - \frac{1}{cos^2(\alpha)})}{cos^2(\alpha)(1 - \frac{1}{sin^2(\alpha)})} = \frac{sin^2(\alpha)(\frac{cos^2(\alpha) - 1}{cos^2(\alpha)})}{cos^2(\alpha)(\frac{sin^2(\alpha) - 1}{sin^2(\alpha)})} = \frac{sin^2(\alpha)(\frac{-sin^2(\alpha)}{cos^2(\alpha)})}{cos^2(\alpha)(\frac{-cos^2(\alpha)}{sin^2(\alpha)})} = \frac{sin^4(\alpha)}{cos^4(\alpha)} = tg^4(\alpha)$. *Перевод: (sin^2 α - tg^2 α) / (cos^2 α - ctg^2 α) = (sin^2(α) - sin^2(α)/cos^2(α)) / (cos^2(α) - cos^2(α)/sin^2(α)) = (sin^2(α)(1 - 1/cos^2(α))) / (cos^2(α)(1 - 1/sin^2(α))) = (sin^2(α)((cos^2(α) - 1)/cos^2(α))) / (cos^2(α)((sin^2(α) - 1)/sin^2(α))) = (sin^2(α)((-sin^2(α))/cos^2(α))) / (cos^2(α)((-cos^2(α))/sin^2(α))) = sin^4(α) / cos^4(α) = tg^4(α)*