If csc(-x)=1/sin(-x)=-1/sin(x)=-6, and sin(x)=-1/6. Sine is negative in quadrants 3 and 4.
Forgetting for a moment the sign, we have a right-angled triangle where (opposite)/(hypotenuse)=sin(x), then the hypotenuse is represented by 6 and opposite by 1. The remaining side has length sqrt(36-1)=sqrt(35), so cos(x)=sqrt(35)/6. Cosine is positive in quadrants 1 and 4. The common quadrant 4 applies for negative sine and positive cosine. In quadrant 4 tangent is negative, so tan(x)=-1/sqrt(35).
And sec(x)=1/cos(x)=6/sqrt(35)=6sqrt(35)/35; and cot(x)=-sqrt(35).