To find the value of tg(x) at x = -1, we first need to find the value of F(x) at x = -1 using the given function F(x) = 3x + 4/x^3.

F(-1) = 3*(-1) + 4/(-1)^3
F(-1) = -3 + 4/(-1)
F(-1) = -3 - 4
F(-1) = -7

Now, we need to find tg(x):
tg(x) = sin(x)/cos(x)

So, we need to find sin(x) and cos(x) at x = -1.

sin(-1) = sin(-pi/4) = -sqrt(2)/2
cos(-1) = cos(-pi/4) = sqrt(2)/2

Therefore, tg(-1) = sin(-1)/cos(-1) = (-sqrt(2)/2)/(sqrt(2)/2) = -1

So, tg(x) at x = -1 is -1.