Sinx+2cosx=0, then sinx=-2, sinx/cosx=tanx=-2.

cos2x=(conx)^2-(sinx)^2

sin2x=2sinxconx

1=(conx)^2+(sinx)^2

The equation becomes = [(conx) 2-(sinx) 2-2 sinx conx]/[2 (conx) 2+(sinx) 2]

When (conx)^2) is added up and down at the same time, [1-(tanx) 2-2 tanx]/[2+(tanx) 2] =1/6 is obtained.

If you don't understand, you can look at the picture.