Question

Решить уравнения: 1. cos2x-sin(п/2+x)+1=0 2. cos2x+sin(п/2-x)+1=0

Answer 1

$1. \ \ cos2x-sin\left ( \frac{\pi}{2}+x \right )+1=0\Leftrightarrow cos2x-cosx+1=0\\2cos^2x-cosx=0\Leftrightarrow cosx\left ( 2cosx-1 \right )=0\\cosx=0\Rightarrow x=\frac{\pi}{2}+\pi k,k\in \mathbb{Z}\\2cosx=1\Rightarrow x=\pm \frac{\pi}{3}+2\pi k,k\in \mathbb{Z}$

$2. \ \ cos2x+sin\left ( \frac{\pi}{2}-x \right )+1=0\Leftrightarrow cos2x+cosx+1=0\\2cos^2x+cosx=0\Leftrightarrow cosx\left ( 2cosx+1 \right )=0\\cosx=0\Rightarrow x=\frac{\pi}{2}+\pi k,k\in \mathbb{Z}\\2cosx=-1\Rightarrow x=\pm \frac{2\pi}{3}+2\pi k,k\in \mathbb{Z}$