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а)![sin4x+cos^22x=2 \
2sin2xcos2x+cos^22x-2cos^22x-2sin^22x=0 \
2sin^22x-2sin2xcos2x+cos^22x=0 |:cos^22x neq 0 \
2tg^22x-2tg2x+1=0 \
tg2x=t \
2t^2-2t+1=0 \
D textless 0](https://tex.z-dn.net/?f=sin4x%2Bcos%5E22x%3D2+%5C+%0A2sin2xcos2x%2Bcos%5E22x-2cos%5E22x-2sin%5E22x%3D0+%5C+%0A2sin%5E22x-2sin2xcos2x%2Bcos%5E22x%3D0+%7C%3Acos%5E22x+neq+0+%5C+%0A2tg%5E22x-2tg2x%2B1%3D0+%5C+%0Atg2x%3Dt+%5C+%0A2t%5E2-2t%2B1%3D0+%5C+%0AD+textless++0%0A "sin4x+cos^22x=2 \
2sin2xcos2x+cos^22x-2cos^22x-2sin^22x=0 \
2sin^22x-2sin2xcos2x+cos^22x=0 |:cos^22x neq 0 \
2tg^22x-2tg2x+1=0 \
tg2x=t \
2t^2-2t+1=0 \
D textless 0")
Действительных корней нет.
б)
![cos^4( frac{3x}{2} )-sin^4( frac{3x}{2} )= frac{1}{2} \
(cos^2( frac{3x}{2} )-sin^2( frac{3x}{2} ))(cos^2( frac{3x}{2} )+sin^2( frac{3x}{2} ))= frac{1}{2} \
cos^2( frac{3x}{2} )-sin^2( frac{3x}{2} )= frac{1}{2} \
cos^2( frac{3x}{2} )-(1-cos^2( frac{3x}{2} ))= frac{1}{2} \
2cos^2( frac{3x}{2} )= frac{3}{2} \
cos^2( frac{3x}{2} )= frac{3}{4} \
cos( frac{3x}{2} )= frac{ sqrt{3} }{2} \
frac{3x}{2} =+- frac{ pi }{6} +2 pi n \](https://tex.z-dn.net/?f=cos%5E4%28+frac%7B3x%7D%7B2%7D+%29-sin%5E4%28+frac%7B3x%7D%7B2%7D+%29%3D+frac%7B1%7D%7B2%7D++%5C+%0A%28cos%5E2%28+frac%7B3x%7D%7B2%7D+%29-sin%5E2%28+frac%7B3x%7D%7B2%7D+%29%29%28cos%5E2%28+frac%7B3x%7D%7B2%7D+%29%2Bsin%5E2%28+frac%7B3x%7D%7B2%7D+%29%29%3D+frac%7B1%7D%7B2%7D++%5C+%0Acos%5E2%28+frac%7B3x%7D%7B2%7D+%29-sin%5E2%28+frac%7B3x%7D%7B2%7D+%29%3D+frac%7B1%7D%7B2%7D++%5C+%0Acos%5E2%28+frac%7B3x%7D%7B2%7D+%29-%281-cos%5E2%28+frac%7B3x%7D%7B2%7D+%29%29%3D+frac%7B1%7D%7B2%7D++%5C+%0A2cos%5E2%28+frac%7B3x%7D%7B2%7D+%29%3D+frac%7B3%7D%7B2%7D++%5C+%0Acos%5E2%28+frac%7B3x%7D%7B2%7D+%29%3D+frac%7B3%7D%7B4%7D++%5C+%0Acos%28+frac%7B3x%7D%7B2%7D+%29%3D+frac%7B+sqrt%7B3%7D+%7D%7B2%7D+%5C+%0A++frac%7B3x%7D%7B2%7D+%3D%2B-+frac%7B+pi+%7D%7B6%7D+%2B2+pi+n+%5C+%0A "cos^4( frac{3x}{2} )-sin^4( frac{3x}{2} )= frac{1}{2} \
(cos^2( frac{3x}{2} )-sin^2( frac{3x}{2} ))(cos^2( frac{3x}{2} )+sin^2( frac{3x}{2} ))= frac{1}{2} \
cos^2( frac{3x}{2} )-sin^2( frac{3x}{2} )= frac{1}{2} \
cos^2( frac{3x}{2} )-(1-cos^2( frac{3x}{2} ))= frac{1}{2} \
2cos^2( frac{3x}{2} )= frac{3}{2} \
cos^2( frac{3x}{2} )= frac{3}{4} \
cos( frac{3x}{2} )= frac{ sqrt{3} }{2} \
frac{3x}{2} =+- frac{ pi }{6} +2 pi n \")
![x=+- frac{ pi }{9} + frac{2 pi n}{3} \
cos( frac{3x}{2} )=- frac{ sqrt{3}}{2} \
x=+- frac{5 pi }{9} + frac{2 pi n}{3}](https://tex.z-dn.net/?f=x%3D%2B-+frac%7B+pi+%7D%7B9%7D+%2B+frac%7B2+pi+n%7D%7B3%7D++%5C+%0Acos%28+frac%7B3x%7D%7B2%7D+%29%3D-+frac%7B+sqrt%7B3%7D%7D%7B2%7D++%5C+%0Ax%3D%2B-+frac%7B5+pi+%7D%7B9%7D+%2B+frac%7B2+pi+n%7D%7B3%7D+%0A%0A "x=+- frac{ pi }{9} + frac{2 pi n}{3} \
cos( frac{3x}{2} )=- frac{ sqrt{3}}{2} \
x=+- frac{5 pi }{9} + frac{2 pi n}{3}")
n∈Z
Действительных корней нет.
б)
n∈Z
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