Тригонометрия

[nishka]

[tex]ctg^{2}\alpha-3 = \frac{cos3\alpha }{cos\alpha sin^{2}\alpha}[/tex]

да се докаже тъждеството

[Martin Nikovski]

Преобразуваме дясната страна:
[tex]\frac{cos3\alpha}{cos\alpha sin^2\alpha }=\frac{cos\ \left(2\alpha +\alpha \right)}{cos\alpha sin^2\alpha }=\frac{cos2\alpha cos\alpha-sin2\alpha sin\alpha }{cos\alpha sin^2\alpha }=\frac{cos2\alpha}{sin^2\alpha }-\frac{sin2\alpha}{sin\alpha cos\alpha }=\frac{cos^2\alpha-sin^2\alpha}{sin^2\alpha }-\frac{2sin\alpha cos\alpha}{sin\alpha cos\alpha }=[/tex]
[tex]=\frac{cos^2\alpha}{sin^2\alpha }-1-2=cotg^2\alpha -3[/tex]