Question

Dado sen x = 1/3 e π/2 < x < π, calcule o valor da expressão (imagem)

Answer 1

$\boxed{\sin^2{x}+\cos^2{x}=1}$

$\boxed{\sin^2{x}=1-\cos^2{x}}$

Dessa forma:

$A=\dfrac{\sec^2{x}-1}{\tan^2{x}+1}=\dfrac{\frac{1}{\cos^2{x}}-1}{\frac{\sin^2{x}}{\cos^2{x}}+1}=\dfrac{\frac{1-\cos^2{x}}{\cos^2{x}}}{\frac{\sin^2{x}+\cos^2{x}}{\cos^2{x}}}\ \to$

$A=\dfrac{1-\cos^2{x}}{1}\ \to\ \boxed{A=\sin^2{x}}$

$\sin{x}=\dfrac{1}{3}\ \to\ A=\bigg(\dfrac{1}{3}\bigg)^2\ \to\ \boxed{A=\dfrac{1}{9}}$