[tex]\displaystyle\bf\\\frac{1+tg^{2}\alpha }{1-tg^{2} \alpha } =\frac{1+\frac{Sin^{2}\alpha }{Cos^{2} \alpha } }{1-\frac{Sin^{2}\alpha }{Cos^{2}\alpha } } =\frac{\frac{Cos^{2} \alpha +Sin^{2} \alpha }{Cos^{2} \alpha } }{\frac{Cos^{2} \alpha -Sin^{2} \alpha }{Cos^{2} \alpha } } =\frac{\frac{1 }{Cos^{2} \alpha } }{\frac{Cos^{2} \alpha -Sin^{2} \alpha }{Cos^{2} \alpha } } =\\\\\\=\frac{Cos^{2} \alpha }{Cos^{2} \alpha \cdot(Cos^{2} \alpha -Sin^{2}\alpha ) } =\frac{1}{Cos^{2} \alpha -Sin^{2} \alpha }[/tex]

[tex]\displaystyle\bf\\\frac{1}{Cos^{2} \alpha -Sin^{2} \alpha } =\frac{1}{Cos^{2} \alpha -Sin^{2} \alpha }[/tex]

Что и требовалось доказать .