Question

Como resolver a expressão:

y = (sec x - tg x) (sec x + tg x) / (1 - sen x) (cotg x - cossec x) (cotg x + cossec x) ?

Answer 1

$\begin{array}{c} \textsf{Antes de dar in\'icio, vale lembrar que:}\end{array}$


$\mathsf{\checkmark\quad sec^2x-tg^2x=1}\\\\\\ \mathsf{\checkmark\quad cotg^2x-cossec^2x=-1}$

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$\begin{array}{c} \textsf{Primeiramente vamos resolver os produtos not\'aveis}\end{array}$


$\mathsf{y=\dfrac{(sec~x-tg~x)\cdot (sec~x+tg~x)}{(1-sen~x)\cdot (cotg~x-cossec~x)\cdot (cotg~x+cossec~x)}}\\\\\\ \mathsf{y=\dfrac{sec^2~x-tg^2~x}{(1-sen~x)\cdot (cotg^2~x-cossec^2~x)}}$



$\begin{array}{c} \textsf{Agora com as rela\c{c}\~oes citadas, podemos substituir termos por 1 e -1}\end{array}$

 
$\mathsf{y=\dfrac{1}{(1-sen~x)\cdot (-1)}}\\\\\\ \mathsf{y=\dfrac{1}{(sen~x-1)}}\\\\\\ \boxed{\begin{array}{c} \mathsf{y=(sen~x-1)^{-1}}\end{array}}$



Bons estudos no Brainly! =)