Question

Докажите тождество..
sin^4a-2sin^2a*cos^2a+cos^4а/(sina+cosa)^2 = 1-sin^2a

Answer 1

фото:

$frac{sin^4 alpha -2sin^2 alpha cdot cos^2 alpha +cos^4 alpha }{(sin alpha +cos alpha )^2}=1-sin2alpha$

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$frac{sin^4 alpha -2sin^2 alpha cdot cos^2 alpha +cos^4 alpha }{(sin alpha +cos alpha )^2}=$

$frac{(cos^2alpha-sin^2alpha)^2}{sin^2alpha+2sinalpha cosalpha+cos^2alpha}=$

$frac{(cos2alpha)^2}{1+2sinalpha cosalpha}=frac{(cos2alpha)^2}{1+sin2alpha}=$

$frac{cos^22alpha}{1+sin2alpha}=frac{1-sin^22alpha}{1+sin2alpha}=$

$frac{(1-sin2alpha)(1+sin2alpha)}{1+sin2alpha}=1-sin2alpha$