### Instruction: Find the curve defined by the equation
\[r = 4 \tan \theta \sec \theta.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.### Response: I need to find the shape of the curve that this polar equation describes.
I could try to plot some points, but that might not give me a clear picture of the curve, especially if it has asymptotes or branches.
I could also try to convert the equation to rectangular coordinates by using the identities $x = r \cos \theta$ and $y = r \sin \theta$, and then see if I can recognize the standard form of a conic section.
To do that, I need to simplify the equation first.
I notice that $\tan \theta \sec \theta = \sin \theta / \cos^2 \theta$, so I can rewrite the equation as
\[r = 4 \sin \theta / \cos^2 \theta.\]
Now I can multiply both sides by $r \cos^2 \theta$ to get
\[r^2 \cos^2 \theta = 4 r \sin \theta.\]
Substituting $x = r \cos \theta$ and $y = r \sin \theta$, I get
\[x^2 = 4y.\]
This is the equation of a parabola with vertex at the origin and axis of symmetry along the positive y-axis.