Как се намират котангенсите на половинките на ъглите на

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Как се намират котангенсите на половинките на ъглите на триъгълник

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[tex]cotg{\frac{\alpha}{2}}=\frac{cos{\frac{\alpha}{2}}}{sin{\frac{\alpha}{2}}};[/tex]
[tex]cos\alpha=cos^2{\frac{\alpha}{2}}-sin^2{\frac{\alpha}{2}}=2cos^2{\frac{\alpha}{2}}-1 \Rightarrow cos{\frac{\alpha}{2}}=\pm \sqrt{\frac{1+cos\alpha}{2}};[/tex]
[tex]cos\alpha=cos^2{\frac{\alpha}{2}}-sin^2{\frac{\alpha}{2}}=1-2sin^2{\frac{\alpha}{2}} \Rightarrow sin{\frac{\alpha}{2}}=\pm \sqrt{\frac{1-cos\alpha}{2}};[/tex]
[tex]\alpha \in (0^\circ; 90^\circ) \Rightarrow sin{\frac{\alpha}{2}}=\sqrt{\frac{1-cos\alpha}{2}}; cos{\frac{\alpha}{2}}\sqrt{\frac{1+cos\alpha}{2}} \Rightarrow cotg{\frac{\alpha}{2}}=\sqrt{\frac{1+cos\alpha}{2}.\frac{2}{1-cos\alpha}}=\sqrt{\frac{1+cos\alpha}{1-cos\alpha}}[/tex]

[tex]cotg{\frac{\beta}{2}}=\frac{cos{\frac{\beta}{2}}}{sin{\frac{\beta}{2}}};[/tex]
[tex]cos\beta=cos^2{\frac{\beta}{2}}-sin^2{\frac{\beta}{2}}=2cos^2{\frac{\beta}{2}}-1 \Rightarrow cos{\frac{\beta}{2}}=\pm \sqrt{\frac{1+cos\beta}{2}};[/tex]
[tex]cos\beta=cos^2{\frac{\beta}{2}}-sin^2{\frac{\beta}{2}}=1-2sin^2{\frac{\beta}{2}} \Rightarrow sin{\frac{\beta}{2}}=\pm \sqrt{\frac{1-cos\beta}{2}};[/tex]
[tex]\beta \in (0^\circ; 90^\circ) \Rightarrow sin{\frac{\beta}{2}}=\sqrt{\frac{1-cos\beta}{2}}; cos{\frac{\beta}{2}}\sqrt{\frac{1+cos\beta}{2}} \Rightarrow cotg{\frac{\beta}{2}}=\sqrt{\frac{1+cos\beta}{2}.\frac{2}{1-cos\beta}}=\sqrt{\frac{1+cos\beta}{1-cos\beta}}[/tex]

[tex]cotg{\frac{\gamma}{2}}=\frac{cos{\frac{\gamma}{2}}}{sin{\frac{\gamma}{2}}};[/tex]
[tex]cos\gamma=cos^2{\frac{\gamma}{2}}-sin^2{\frac{\gamma}{2}}=2cos^2{\frac{\gamma}{2}}-1 \Rightarrow cos{\frac{\gamma}{2}}=\pm \sqrt{\frac{1+cos\gamma}{2}};[/tex]
[tex]cos\gamma=cos^2{\frac{\gamma}{2}}-sin^2{\frac{\gamma}{2}}=1-2sin^2{\frac{\gamma}{2}} \Rightarrow sin{\frac{\gamma}{2}}=\pm \sqrt{\frac{1-cos\gamma}{2}};[/tex]
[tex]\gamma \in (0^\circ; 90^\circ) \Rightarrow sin{\frac{\gamma}{2}}=\sqrt{\frac{1-cos\gamma}{2}}; cos{\frac{\gamma}{2}}\sqrt{\frac{1+cos\gamma}{2}} \Rightarrow cotg{\frac{\gamma}{2}}=\sqrt{\frac{1+cos\gamma}{2}.\frac{2}{1-cos\gamma}}=\sqrt{\frac{1+cos\gamma}{1-cos\gamma}}[/tex]