[tex](\sin{\alpha}+\cos{\alpha})^{2}=\sin{\alpha}^{2}+\cos{\alpha}^{2}+2\sin{\alpha}\cos{\alpha}=1+2\sin{\alpha}\cos{\alpha}[/tex]
[tex](\sin{\alpha}-\cos{\alpha})^{2}=\sin{\alpha}^{2}+\cos{\alpha}^{2}-2\sin{\alpha}\cos{\alpha}=1-2\sin{\alpha}\cos{\alpha}[/tex]
[tex]\frac{(\sin{\alpha}+\cos{\alpha})^{2}+(\sin{\alpha}-\cos{\alpha})^{2}}{2ctg\alpha}=\frac{1+2\sin{\alpha}\cos{\alpha}+1-2\sin{\alpha}\cos{\alpha}}{2ctg\alpha}=\frac{2}{2ctg\alpha}=\frac{1}{ctg\alpha}=tg\alpha[/tex]
