tex]...

RoMsTeIn @RoMsTeIn

August 2022 1 9 Report

1. cos²2x - sin²2x = -1

2. 1 + cosx = 2cos [tex] \frac{x}{2} [/tex]

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

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