First, tan(θ) = sin(θ) / cos(θ), so if cos(θ) = 3/5 > 0 and tan(θ) < 0, then it follows that sin(θ) < 0.
Recall the Pythagorean identity:
sin²(θ) + cos²(θ) = 1
Then
sin(θ) = -√(1 - cos²(θ)) = -4/5
and so
tan(θ) = (-4/5) / (3/5) = -4/3
The remaining trig ratios are just reciprocals of the ones found already:
sec(θ) = 1/cos(θ) = 5/3
csc(θ) = 1/sin(θ) = -5/4
cot(θ) = 1/tan(θ) = -3/4
