Decide whether the statement is possible or impossible. \[ \cos \theta<0 \text { and } \sec \theta<0 \] Is the statement possible or impossible? A. The statement is impossible because $\cos \theta$ is always positive. B. The statement is impossible because cosine and secant always have opposite signs. C. The statement is possible because $\sec \theta$ is always negative. D. The statement is possible because cosine and secant always have the same sign. E. The statement is impossible because $\sec \theta$ is always positive. $\mathrm{F}$. The statement is possible because $\cos \theta$ is always negative.
Solution
Step 1 ：The secant of an angle is defined as the reciprocal of the cosine of the angle. Therefore, if the cosine of an angle is negative, its secant will also be negative. Conversely, if the cosine of an angle is positive, its secant will also be positive.
Step 2 ：Let's take an example where \(\theta = 2.0943951023931953\).
Step 3 ：Calculate the cosine of \(\theta\), \(\cos \theta = -0.4999999999999998\).
Step 4 ：Calculate the secant of \(\theta\), \(\sec \theta = -2.000000000000001\).
Step 5 ：The cosine and secant of the angle \(\theta\) are both negative, which confirms that it is possible for both the cosine and secant of an angle to be negative at the same time.
Step 6 ：Final Answer: The statement is possible because cosine and secant always have the same sign. So, the correct option is \(\boxed{\text{(D)}}\).
