19a)
Pornim de la membrul drept și, prin transformări, ajungem la membrul stâng.
[tex]\it 2\cdot \dfrac{c \cdot cosB+b\cdot cosC}{c\cdot sinB+b\cdot sinC} =2\cdot \dfrac{c\cdot\dfrac{c}{a}+b\cdot\dfrac{b}{a}}{c\cdot\dfrac{b}{a}+b\cdot\dfrac{c}{a}} =2\cdot \dfrac{\dfrac{c^2+b^2}{a}}{\dfrac{bc+bc}{a}} =
\\\;\\ \\\;\\
=2\cdot\dfrac{c^2+b^2}{2bc}=\dfrac{b^2+c^2}{bc} = \dfrac{b^2}{bc} +\dfrac{c^2}{bc} =\dfrac{b}{c}+\dfrac{c}{b} =tgB+tgC.[/tex]
b)
[tex]\it b\cdot cosB+c\cdot cosC = b\cdot\dfrac{c}{a} +c\cdot \dfrac{b}{a} = \dfrac{bc}{a} + \dfrac{bc}{a} = \dfrac{2bc}{a} =\dfrac{2abc}{a^2} =
\\\;\\ \\\;\\
=2a\cdot\dfrac{bc}{a\cdot a} =2a\cdot \dfrac{b}{a}\cdot\dfrac{c}{a} = 2a\cdot sinB sinC.[/tex]
