It seems that the text extracted from the image wasn't clear enough to interpret correctly. Based on what I can see from the image preview, the question asks:

For the curve $r = 2\cos^2\theta + 4\tan\theta$, what is the polar subtangent at $\theta = \frac{\pi}{4}$?

I will solve this now.

The polar subtangent is given by the formula: $T = \frac{r}{\frac{dr}{d\theta}}$

1. Differentiate $r = 2\cos^2\theta + 4\tan\theta$ with respect to $\theta$.
2. Evaluate both $r$ and $\frac{dr}{d\theta}$ at $\theta = \frac{\pi}{4}$.
3. Compute the subtangent.

Let's begin by calculating it.The polar subtangent at $\theta = \frac{\pi}{4}$ is:

$T = \frac{r}{\frac{dr}{d\theta}} = \frac{5}{6}$

So, the correct answer is not directly listed, but $5/6$ is the subtangent. If you have any more details to check, feel free to ask.

Here are 5 related questions to expand on this:

1. How is the polar subtangent used in determining curve properties?
2. Can the subtangent ever be negative? If so, under what conditions?
3. How do other coordinate systems (e.g., Cartesian) treat the subtangent concept?
4. What is the geometrical meaning of the subtangent in a polar curve?
5. How does the subtangent relate to the overall behavior of the curve at that point?

Tip: Always verify the formula before starting derivative calculations to ensure proper handling of trigonometric functions!