[tex]\dfrac{\text{tg}\,\alpha}{1-\text{tg}^2\alpha}\cdot \dfrac{\text{ctg}^2\alpha-1}{\text{ctg}\,\alpha} =\dfrac{\text{tg}\,\alpha}{1-\text{tg}^2\alpha}\cdot \dfrac{\text{ctg}^2\alpha-1}{\text{ctg}\,\alpha}\cdot\dfrac{\text{tg}^2\alpha}{\text{tg}^2\alpha}=\\ \\ =\dfrac{\text{tg}\,\alpha}{1-\text{tg}^2\alpha}\cdot \dfrac{\text{tg}^2\alpha\cdot \text{ctg}^2\alpha-\text{tg}^2\alpha}{\text{tg}^2\alpha\cdot \text{ctg}\,\alpha} = \dfrac{\text{tg}\,\alpha}{1-\text{tg}^2\alpha}\cdot \dfrac{1-\text{tg}^2\alpha}{\text{tg}\alpha} =\\ \\ = \boxed{1}[/tex]
