Rozwiązania:
A) Dowód:
[tex]sin2\alpha =sin(\alpha +\alpha )=sin\alpha cos\alpha +cos\alpha sin\alpha =2sin\alpha cos\alpha[/tex]
B) Lemat:
[tex]cos2\alpha =cos(\alpha +\alpha )=cos\alpha *cos\alpha -sin\alpha *sin\alpha =cos^{2}\alpha-sin^{2} \alpha[/tex]
Dowód:
[tex]tg2\alpha =\frac{sin2\alpha }{cos2\alpha }=\frac{2sin\alpha cos\alpha }{cos^{2}\alpha -sin^{2}\alpha } =\frac{2sin\alpha cos\alpha }{cos^{2}\alpha (1-\frac{sin^{2}\alpha }{cos^{2}\alpha }) }=\frac{2sin\alpha }{cos\alpha (1-tg^{2}\alpha ) }=\frac{2tg\alpha }{1-tg^{2} \alpha }[/tex]
[tex]q.e.d.[/tex]
