Find sin(2x), cos(2x), and tan(2x) from the given information. csc(x) = 8, tan(x) < 0

If csc x = 1  sin x = 1/8  and also as tan x < 0  x must be in the second quadrant.

If hypotenuse = 8 and opposite side = 1 then adjacent side  =  -sqrt63
(negative because it is in second quadrant)
so cos x = -sqrt63/8

sin 2x  = 2 sin x cos x =  2 * 1/8 * -sqrt63 / 8 = -sqrt63/32   Answer

cos 2x = cos^2 x - sin^2 x =  63/64 - 1/64 =  31/32  Answer

tan 2x = sin 2x / cos 2x =  -sqrt63 / 32 * 32 / 31  =  - sqrt63/ 31  Answer