Question

PROVAR QUE (1/1+tg²(alfa))+sen²(alfa)=1.

Answer 1

Nos sabemos que

$\mathsf{tg\alpha = \dfrac{sen\alpha}{cos\alpha} \ e \ sen^2\alpha + cos^2\alpha = 1 }$

Então

$\mathsf{ \dfrac{1}{1 + tg^2\alpha} = \dfrac{1}{1 + \frac{sen^2\alpha}{cos^2\alpha}} = \dfrac{1}{\frac{sen^2\alpha + cos^2\alpha}{cos^2\alpha}} = \dfrac{1}{\frac{1}{cos^2\alpha}} = cos^2\alpha}$

Portanto

$\mathsf{\dfrac{1}{1 + tg^2\alpha} + sen^2\alpha = cos^2\alpha + sen^2\alpha = 1}$