Question

Mostre a seguinte idêntidade trigonométrica:

(tgx . cotgx) . (secx - cosx) . (cossecx - senx) = 1

Answer 1

$\boxed{\begin{array}{l}\rm (tg(x)\cdot cotg(x))\cdot(sec(x)-cos(x))\cdot(cossec(x)-sen(x))=1\\\rm\bigg(tg(x)\cdot\dfrac{1}{tg(x)}\bigg)\cdot\bigg(\dfrac{1}{cos(x)}-cos(x)\bigg)\cdot\bigg(\dfrac{1}{sen(x)}-sen(x)\bigg)\\\\\rm1\cdot\bigg(\dfrac{1-cos^2(x)}{cos(x)}\bigg)\cdot\bigg(\dfrac{1-sen^2(x)}{sen(x)}\bigg)\\\\\rm 1\cdot\dfrac{sen^2(x)}{cos(x)}\cdot\dfrac{cos^2(x)}{sen(x)}\end{array}}$

$\boxed{\begin{array}{l}\rm\dfrac{(sen(x)\cdot cos(x))^2}{sen(x)\cdot cos(x)}=sen(x)\cdot cos(x)\\\rm A~identidade~fornecida~no~enunciado\\\rm\acute e~falsa.\end{array}}$