Question

1. cos²2x - sin²2x = -1

2. 1 + cosx = 2cos $frac{x}{2}$

Answer 1

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