Question

Demonstre a identidade trigonométrica cos x . tg x . cossec x= 1

Answer 1

Vamos lembrar que tg(x) é o quociente entre sen(x) e cos(x) e que cossec(x) é o inverso do sen(x).

$\boxed{tg(x)~=~\dfrac{sen(x)}{cos(x)}}~~~~\boxed{cossec(x)~=~\dfrac{1}{sen(x)}}$

Substituindo tg(x) e cossec(x) na expressão "cos(x).tg(x).cossec(x)", fica:

$cos(x)\cdot tg(x)\cdot cossec(x)~=~cos(x)\cdot \dfrac{sen(x)}{cos(x)}\cdot \dfrac{1}{sen(x)}\\\\\\cos(x)\cdot tg(x)\cdot cossec(x)~=~\dfrac{cos(x)\cdot sen(x)\cdot 1}{cos(x)\cdot sen(x)}\\\\\\Dividindo~numerador~e~denominador~por~sen(x)\\\\\\cos(x)\cdot tg(x)\cdot cossec(x)~=~\dfrac{cos(x)\cdot sen(x)\cdot 1~~~~\div~sen(x)}{cos(x)\cdot sen(x)~~~~~~~~\div~sen(x)}\\\\\\cos(x)\cdot tg(x)\cdot cossec(x)~=~\dfrac{cos(x)\cdot 1\cdot 1}{cos(x)\cdot 1\cdot 1}\\\\\\Dividindo~numerador~e~denominador~por~cos(x)$

$cos(x)\cdot tg(x)\cdot cossec(x)~=~\dfrac{cos(x)\cdot 1\cdot 1~~~~\div~cos(x)}{cos(x)\cdot 1~~~~~~~~\div~cos(x)}\\\\\\cos(x)\cdot tg(x)\cdot cossec(x)~=~\dfrac{1\cdot 1\cdot 1}{1\cdot 1}\\\\\\\boxed{cos(x)\cdot tg(x)\cdot cossec(x)~=~1}\\\\\\\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$