In which quadrant does $\theta$ lie if the following statements are true: \[ \csc \theta<0 \text { and }(\csc \theta)(\cos \theta)<0 \]

Step 1 ：The cosecant function, \(\csc \theta\), is negative in the third and fourth quadrants.

Step 2 ：The cosine function, \(\cos \theta\), is negative in the second and third quadrants.

Step 3 ：Therefore, if the product of \(\csc \theta\) and \(\cos \theta\) is negative, then one of the functions must be positive and the other must be negative.

Step 4 ：This can only occur in the third quadrant, where \(\csc \theta\) is negative and \(\cos \theta\) is positive.

Step 5 ：Final Answer: \(\theta\) lies in the \(\boxed{\text{third quadrant}}\).