Question

Sen x = 1/3 e pi/2 < x < pi calcule o valor de y= cotg x - sec x

Answer 1

$\Large\boxed{\begin{array}{l}\sf sen(x)=\dfrac{1}{3}\implies cossec(x)=3\\\sf cossec^2(x)=9\\\sf 1+cotg^2(x)=cossec^2(x)\\\sf 1+cotg^2(x)=9\\\sf cotg^2(x)=9-1\\\sf cotg^2(x)=8\\\sf cotg(x)=-\sqrt{8}=-2\sqrt{2}\\\sf cotg(x)=\dfrac{cos(x)}{sen(x)}\\\\\sf -2\sqrt{2}=\dfrac{cos(x)}{\dfrac{1}{3}}\\\\\sf cos(x)=-\dfrac{2\sqrt{2}}{3}\implies sec(x)=-\dfrac{3}{2\sqrt{2}}=-\dfrac{3\sqrt{2}}{4}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf y=cotg(x)-sec(x)\\\\\sf y=-2\sqrt{2}-\bigg(-\dfrac{3\sqrt{2}}{4}\bigg)\\\\\sf y=-2\sqrt{2}+\dfrac{3\sqrt{2}}{4}\\\\\sf y=\dfrac{-8\sqrt{2}+3\sqrt{2}}{4}\\\\\sf y=-\dfrac{5\sqrt{2}}{4}\end{array}}$