Question

Considerando que senx= 2/3 e x pertence ao segundo quadrante, o valor de tgx+cotgx/secx+cossecx é:

a) -senx d)-3(2+ raiz de 5)
b)raiz de 5/3 e) -6 + raiz de 5
c)cos²x

Answer 1

$\frac{tagx+cotagx}{secx+cossecx} = \frac{ \frac{(senx)^2+(cosx) ^2}{sen x cosx} }{ \frac{1}{cosx} + \frac{1}{senx} } =\frac{ \frac{1}{sen x cosx} }{ \frac{1}{cosx} + \frac{1}{senx} } =\frac{ \frac{1}{sen x cosx} }{ \frac{senx+cosx}{cosxsex}} \\ \frac{1}{senx+cosx} = \frac{1}{ \frac{2}{3} - \sqrt{1-( \frac{2}{3} )^2} } = \frac{1}{ \frac{2- \sqrt{5} }{3} } = \frac{3}{2- \sqrt{5} }} \frac{2+ \sqrt{5} }{2+\sqrt{5}} \\ \frac{3(2+ \sqrt{5}) }{-1} = - 3 (2+\sqrt{5})$