[Analysis]
This problem tests the understanding of trigonometric function signs, the representation of quadrant angles, and the application of inequality properties. By analyzing the given conditions, we can determine the quadrant in which $θ$ lies, and subsequently solve for $\frac{θ}{2}$.

[Solution]
$|cos⁡θ|=cos⁡θ$, which means $θ$ is in the first, fourth quadrant or on the positive half of the $x$-axis.
$|tan⁡θ|=-tan⁡θ$, which means $θ$ is in the second, fourth quadrant or on the $x$-axis.

Therefore, $θ$ is in the fourth quadrant or on the positive half of the $x$-axis.
$k⋅360∘+270∘<θ≤k⋅360∘+360∘$, where $k∈\mathbb{Z}$.

Hence, $k⋅180∘+135∘<\frac{θ}{2}≤k⋅180∘+180∘$, where $k∈\mathbb{Z}$.

Let $k=2n$, where $n∈\mathbb{Z}$,
Then $n⋅360∘+135∘<\frac{θ}{2}≤n⋅360∘+180∘$, which is in the second quadrant or on the negative half of the $x$-axis.

Let $k=2n+1$, where $n∈\mathbb{Z}$,
Then $n⋅360∘+315∘<\frac{θ}{2}≤n⋅360∘+360∘$, which is in the fourth quadrant or on the positive half of the $x$-axis.

Thus, the answer is $\boxed{D}$.