Question

Доказать тождество/Довести тотожність
(1+tga*tgb)/(1-tga*tgb) = cos(a-b)/cos(a+b)

Answer 1

$\displaystile\bf\\\dfrac{1+tg\alpha \cdot tg\beta }{1-tg\alpha \cdot tg\beta }=\dfrac{1+\dfrac{Sin\alpha }{Cos\alpha }\cdot\dfrac{Sin\beta }{Cos\beta } }{1-\dfrac{Sin\alpha }{Cos\alpha } \cdot\dfrac{Sin\beta }{Cos\beta } } =\\\\\\=\dfrac{(Cos\alpha Cos\beta +Sin\alpha Sin\beta )\cdot Cos\alpha Cos\beta }{(Cos\alpha Cos\beta -Sin\alpha Sin\beta )\cdot Cos\alpha Cos\beta } =\dfrac{Cos(\alpha -\beta )}{Cos(\alpha +\beta )}$

$\displaystyle\bf\\\dfrac{Cos(\alpha -\beta )}{Cos(\alpha +\beta )} =\dfrac{Cos(\alpha -\beta )}{Cos(\alpha +\beta )}$

Тождество доказано