Question

sendo sen x = 1/2 com 0 < x < pi/2 qual é o valor da expressao y = 2 tg² x + sec² x

Answer 1

$\mathrm{\sin{x}=\dfrac{1}{2}\ \to\ \sin^2{x}=\bigg(\dfrac{1}{2}\bigg)^2\ \to\ \sin^2{x}=\dfrac{1}{4}}\\\\ \mathrm{\sin^2{x}+cos^2{x}=1\ \to\ \cos^2{x}=1-\sin^2{x}\ \to\ \cos^2{x}=1-\dfrac{1}{4}\ \to}\\\\ \mathrm{\to\ \cos^2{x}=\dfrac{4}{4}-\dfrac{1}{4}\ \to\ \cos^2{x}=\dfrac{3}{4}}$

$\mathrm{y=2\tan^2{x}+\sec^2{x}=2\dfrac{\sin^2{x}}{\cos^2{x}}+\dfrac{1}{\cos^2{x}}=\dfrac{2\sin^2{x}+1}{\cos^2{x}}=}\\\\ \mathrm{=\dfrac{2\dfrac{1}{4}+1}{\dfrac{3}{4}}=\dfrac{\dfrac{1}{2}+\dfrac{2}{2}}{\dfrac{3}{4}}=\dfrac{\dfrac{3}{2}}{\dfrac{3}{4}}=\dfrac{3}{2}.\dfrac{4}{3}=\dfrac{4}{2}=2\ \to\ \boxed{\boxed{\mathbf{y=2}}}}$