Question

100 баллов.
Упростите выражение:
cos(-2α) + (2sin(π-2α)) / (ctg+ctg(π/2+α))

Решите уравнение:
2tgx+3=tg(1.5π+x)

Answer 1

$\displaystyle \cos(-2 \alpha )+ \frac{2\sin( \pi -2\alpha )}{ctg \alpha +ctg( \frac{\pi}{2} +\alpha )} =\cos2\alpha + \frac{2\sin2\alpha }{ctg\alpha -tg\alpha } =\\ \\ =\cos2\alpha + \frac{2\sin2\alpha \cos\alpha \sin\alpha }{\cos^2\alpha -\sin^2\alpha } =\cos2\alpha +\frac{\sin^22\alpha }{\cos2\alpha } =\cos2\alpha +tg2\alpha \sin2\alpha$

$2tgx+3=tg(1.5 \pi +x)\\ 2tgx+3=-tgx\\ 3tgx=-3\\ tgx=-1\\ \boxed{x=- \frac{\pi}{4} +\pi n,n \in \mathbb{Z}}$