find sin theta if cot theta = -4 and cos theta <0

\cot \theta = -4

This means:

\frac{1}{\tan\theta} = -4

and

\tan\theta = -\frac14

In what quadrant are cos and tan negative? When you found out this, make a quick sketch of a right angled triangle with angle theta, adjacent side 4, opposite side 1. Then use Pythagoras' Theorem to find the length of the hypotenuse.

Then, \sin\theta = \frac{opp}{hyp}

The sign of your answer will depend on the quadrant you chose, and whether sin is positive or negative in that particular quadrant.