Udowodnij tożsamość

[SPORE6]

Udowodnić tożsamość \(\frac{1-2sin^2 \alpha }{1+sin2 \alpha } = \frac{1-tg \alpha }{1+tg \alpha}\)

[kerajs]

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