Question

Докажите тождество плз. 4cos^4a-2cos2a-1/2cos4a=3/2

Answer 1

$4cos^4alpha -2cosbig(2alpha big)-dfrac 12cdot cos big(4alpha big)=dfrac 32\\Big(2cos^2alphaBig)^2-2cosbig(2alpha big)-dfrac 12cdot Big(2cos^2 big(2alpha big)-1Big)=dfrac 32\\Big(cosbig(2alphabig)+1Big)^2-2cosbig(2alpha big)-cos^2 big(2alpha big)+dfrac 12=dfrac 32\\cos^2big(2alphabig)+2cosbig(2alphabig)+1-2cosbig(2alpha big)-cos^2 big(2alpha big)+dfrac 12=dfrac 32\\1+dfrac 12=dfrac 32\\dfrac32=dfrac 32~~~~blacksquare$

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Использованы формулы

$cosbig(2alpha big)=2cos^2alpha -1\\2cos^2alpha =cosbid(2alpha big)+1\\big(a+bbig)^2=a^2+2ab+b^2$