[tex]\sec\theta=\dfrac1{\cos\theta}>0\implies\cos\theta>0[/tex]
This happens in the first and fourth quadrants.
Since cosine is positive,
[tex]\cot\theta=\dfrac{\cos\theta}{\sin\theta}<0\implies\sin\theta<0[/tex]
and this happens in the third and fourth quadrants.
This means both conditions can only occur in the fourth quadrant.
