tan^2(x) + 1 = sec^2(x)

Given sin(x) = 1/4, we can find cos(x) using the Pythagorean identity: sin^2(x) + cos^2(x) = 1
(1/4)^2 + cos^2(x) = 1
1/16 + cos^2(x) = 1
cos^2(x) = 1 - 1/16
cos^2(x) = 15/16
cos(x) = ±√(15)/4

Now that we have sin(x) and cos(x), we can find tan(x):
tan(x) = sin(x) / cos(x)
tan(x) = (1/4) / (±√(15)/4)
tan(x) = 1/√(15) or -1/√(15)

Therefore, tg(x) = 1/√(15) or -1/√(15)