\[\sin{2a} = \frac{2tg\ a}{1 + tg^{2}a};\ \ \ \ \ \]
\[\ a \neq \frac{\pi}{2} + \pi n\]
\[\frac{2\sin a \cdot \cos a}{\sin^{2}a + \cos^{2}a} = \frac{2tg\ a}{1 + tg^{2}a}\]
\[\frac{\frac{2\sin a \cdot \cos a}{\cos^{2}a}}{\frac{\sin^{2}a}{\cos^{2}a} + \frac{\cos^{2}a}{\cos^{2}a}} = \frac{2tg\ a}{1 + tg^{2}a}\]
\[\frac{2tg\ a}{1 + tg^{2}a} = \frac{2tg\ a}{1 + tg^{2}a}\]