Question

sabendo que cosx=1/2 , calcule o valor da expressão €=sec²x-1/tg²x+1​

Answer 1

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

$\sin {}^{2} (x) +\left(\dfrac{1}{2}\right)^2 = 1 \\ \sin {}^{2} (x) + \dfrac{1}{4} = 1 \\ \sin {}^{2} (x) = 1 - \frac{ 1}{4} \\ \sin {}^{2} (x) = \frac{4 - 1}{4} \\ \sin {}^{2} (x) = \frac{3}{4} \\ \sin(x) = \pm\sqrt{ \frac{3 }{4} } \\ \sin(x) = \pm \frac{ \sqrt{3} }{2}$

$\bigstar E = \frac{ \sec {}^{2} (x) - 1}{ \tan {}^{2} (x) + 1 } \bigstar \\ \\ E = \frac{ \frac{1}{ \cos {}^{2} (x) } - 1 }{ \frac{ \sin {}^{2} (x) }{ \cos {}^{2} (x) } + 1} \\ \\ E = \frac{ \frac{1}{( \frac{1}{2} ) {}^{2} } - 1 }{ \frac{ (\frac{ \sqrt{3} }{2} ) {}^{2} }{( \frac{1}{2}) {}^{2} } + 1} \\ \\ E = \frac{ \frac{1}{ \frac{1}{4} } - 1}{ \frac{ \frac{3}{4} }{ \frac{1}{4} } + 1} \\ \\ E = \frac{ \frac{1}{1} . \frac{4}{1} - 1}{ \frac{3}{4}. \frac{4}{1} + 1} \\ \\ E = \frac{ 4 - 1}{ \frac{12}{4} + 1} \\ \\ E = \frac{3}{3 + 1} \\ \\ \boxed{E = \frac{3}{4}}$