Question

Решите уравнение!!!Очень сложное!Или подскажите, как решать)

Answer 1

$cos^47x - sin^47x = sin2x\\\\ (cos^27x - sin^27x)(cos^27x + sin^27x) = sin2x\\\\ cos^27x - sin^27x = sin2x\\\\ cos^27x - (1 - cos^27x) = sin2x\\\\ 2cos^2 7x - 1 = sin2x\\\\ cos14x - sin2x = 0\\\\ cos(8x + 6x) - sin(8x - 6x) = 0\\\\ cos8xcos6x - sin8xsin6x - (sin8xcos6x - cos8xsin6x) = 0\\\\ cos8x(cos6x + sin6x) - sin8x(sin6x + cos6x) = 0\\\\ (cos6x + sin6x)(cos8x - sin8x) = 0\\\\ cos6x + sin6x = 0$

$\sqrt{2}sin(\frac{\pi}{4} + 6x) = 0\\\\ sin(\frac{\pi}{4} + 6x) = 0\\\\ \frac{\pi}{4} + 6x = \pi n, n \in Z\\\\ 6x = \pi n - \frac{\pi}{4}, n \in Z\\\\ \boxed{x = \frac{\pi n}{6}- \frac{\pi}{24}, n \in Z}\\\\ cos8x - sin8x = 0\\\\ \sqrt{2}sin(\frac{\pi}{4} - 8x) = 0\\\\ sin(\frac{\pi}{4} - 8x) = 0\\\\ \frac{\pi}{4} - 8x = \pi n, n \in Z\\\\ -8x = \pi n - \frac{\pi}{4}, n \in Z\\\\ \boxed{x = -\frac{\pi n}{8}+ \frac{\pi}{32}, n \in Z}$