the question is:
given sec x = 4, tan x < 0, find sin(2x), cos(2x)and tan (2x)
we can see that secx > 0 and tanx < 0,
so x must be in quadrants IV
draw a right-angled triangle in that quadrant where
x = 1, y = -4, and r = √17
sinx = -4/√17
cosx = 1/√17
tanx = -4/1 = -4
you should know that:
sin(2x) = 2sinxcos = 2((-4/√17)(1/√17) = -8/17
cos(2x) = cos^2x - sin^2x = 1/17 - 16/17 = -15/17
tan(2x) = sin(2x)/cos(2x) = (-8/17)/(-15/17) = 8/15