Question

1. cos²2x - sin²2x = -1 2. 1 + cosx = 2cos

Answer 1

$cos^22x-sin^22x=-1\\cos4x=-1\\4x= \pi + \pi n, n\in Z\\x= \pi /4+ \pi n/4, n\in Z\\\\1+cosx=2cos \frac{x}{2} \\\\cos^2 \frac{x}{2}-sin^2 \frac{x}{2}-2cos \frac{x}{2}+1=0\\\\cos^2 \frac{x}{2}-1+cos^2 \frac{x}{2}-2cos \frac{x}{2}+1=0\\\\2cos^2 \frac{x}{2}-2cos \frac{x}{2}=0\\\\ 2cos \frac{x}{2}(cos \frac{x}{2}-1)=0\\\\cos \frac{x}{2}=0\\\\ \frac{x}{2}= \pi /2+2 \pi n, n\in Z\\x= \pi +4 \pi n, n\in Z\\\\cos \frac{x}{2}-1=0\\\\cos \frac{x}{2}=1\\\\ \frac{x}{2}=2 \pi n, n\in Z\\x=4 \pi n, n\in Z$