Question

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

Answer 1

фото:

$\frac{sin^4 \alpha -2sin^2 \alpha \cdot cos^2 \alpha +cos^4 \alpha }{(sin \alpha +cos \alpha )^2}=1-sin2\alpha$

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

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

$\frac{(cos2\alpha)^2}{1+2sin\alpha cos\alpha}=\frac{(cos2\alpha)^2}{1+sin2\alpha}=$

$\frac{cos^22\alpha}{1+sin2\alpha}=\frac{1-sin^22\alpha}{1+sin2\alpha}=$

$\frac{(1-sin2\alpha)(1+sin2\alpha)}{1+sin2\alpha}=1-sin2\alpha$