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

Step 1 ：The cosine function is positive in the first and fourth quadrants.

Step 2 ：The tangent function is negative in the second and fourth quadrants.

Step 3 ：Therefore, if both conditions are true, \(\theta\) must lie in the fourth quadrant.

Step 4 ：\(\theta\) lies in the \(\boxed{fourth}\) quadrant.