[tex]\dfrac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha } +\dfrac{sin\alpha +cos\alpha }{sin\alpha -cos\alpha } =\dfrac{( sin\alpha -cos\alpha)\cdot ( sin\alpha -cos\alpha)+(sin\alpha +cos\alpha)\cdot ( sin\alpha +cos\alpha) }{ ( sin\alpha -cos\alpha)\cdot ( sin\alpha +cos\alpha)} =\dfrac{( sin\alpha -cos\alpha)^{2} + ( sin\alpha -cos\alpha)^{2} }{sin^{2} \alpha -cos^{2} \alpha } =[/tex]
[tex]=\dfrac{sin^{2} \alpha -2sin\alpha cos\alpha +cos^{2} \alpha +sin^{2} \alpha +2sin\alpha cos\alpha +cos^{2} \alpha }{sin^{2} \alpha -cos^{2} \alpha } =\dfrac{1+1}{sin^{2} \alpha -cos^{2} \alpha } =\dfrac{2}{sin^{2} \alpha -cos^{2} \alpha }=\dfrac{2}{-cos2\alpha } =-\dfrac{2}{cos2\alpha }[/tex]
korzystam ze wzorów:
sin²α + cos²α = 1
cos2α = cos²α - sin²α ⇒ sin²α - cos²α = - cos2α
( a - b ) × ( a + b ) = a² - b²
