Question

Упростить выражение ((sina-cosa)^2-1)/((sin^2)a-(cos^2)a-1

Answer 1

$\frac{(\sin \alpha - \cos \alpha)^2 -1}{\sin^2 \alpha - \cos^2 \alpha -1}=\frac{\sin^2 \alpha -2 \sin \alpha \cos \alpha +\cos^2 \alpha -1}{-1+\sin^2 \alpha - \cos^2 \alpha}=\frac{\sin^2 \alpha +\cos^2 \alpha -1 -2 \sin \alpha \cos \alpha}{-1+\sin^2 \alpha - \cos^2 \alpha}=\\ \\ = \frac{1 -1 -2 \sin \alpha \cos \alpha}{-(1-\sin^2 \alpha) - \cos^2 \alpha}=\frac{-2\sin \alpha \cos \alpha}{-\cos^2 \alpha - \cos^2 \alpha}=\frac{-2\sin \alpha \cos \alpha}{-2\cos^2 \alpha }=\frac{\sin \alpha}{\cos \alpha}=$$=tg \, \alpha$

Answer 2

[(sina-cosa)²-1]/[sin²a-cos²a-1]=(sin²a-2sinacosa+cos²a-1)/(sin²a-cos²a-sin²a-cos²a)=
=(1-2sinacosa-1)/-2cos²a=2sinacosa/2cos²a=sina/cosa=tga