Question

Demonstrati identitatea:
(sin^4α + cos^4α-1)/ (sin^6α +cos^6 -1) = 2/3

Answer 1

Salut,

$\dfrac{sin^4\alpha+cos^4\alpha-1}{sin^6\alpha +cos^6\alpha-1}=\dfrac{(sin^2\alpha+cos^2\alpha)^2-2\cdot sin^2\alpha\cdot sin^2\alpha-1}{(sin^2\alpha+cos^2\alpha)(sin^4\alpha-sin^2\alpha\cdot cos^2\alpha+cos^4\alpha)-1}=\\\\=\dfrac{1^2-2\cdot sin^2\alpha\cdot sin^2\alpha-1}{1\cdot[(sin^2\alpha+cos^2\alpha)^2-2\cdot sin^2\alpha\cdot sin^2\alpha-sin^2\alpha\cdot sin^2\alpha]-1}=\\\\=\dfrac{-2\cdot sin^2\alpha\cdot sin^2\alpha}{1^2-3\cdot sin^2\alpha\cdot sin^2\alpha-1}=\dfrac{-2}{-3}=\dfrac{2}3,\;ceea\;ce\;trebuia\;demonstrat.$

Green eyes.