If sinθ < 0 and cotθ > 0, then θ lies in the:

sinθ < 0 restricts θ to the third or fourth quadrant. cotθ = cosθ/sinθ is positive only when sine and cosine share a sign; in the fourth quadrant cosθ > 0 while sinθ < 0, so cotθ < 0 there. Only the third quadrant, where both are negative, satisfies both conditions.