The problem in the image can be interpreted and solved as follows:

1. Problem Interpretation:

   • Given: A parallelogram $BCDE$ where $2(OA) = CA$ and $2(OB) = DC$.
   • To Find: $\cos \theta$ and $\tan \theta$.

2. Key Details:

   • $O$ is the origin or a point where segments are originating.
   • $A, B, C, D, E$ are points defining the geometry of the problem, with $OA$ and $OB$ being specific segments whose lengths are used to establish proportional relationships.

3. Geometry of the Problem:

   • $BCDE$ is a parallelogram, meaning opposite sides are equal and parallel.
   • By the problem's statement $2(OA) = CA$, we infer $CA = 2 \cdot OA$.
   • Similarly, $DC = 2 \cdot OB$.

4. Analysis:

   • From these conditions, we can infer that $\triangle OAC$ and $\triangle OBD$ are right triangles (if we assume right angles due to the parallel nature and segments).
   • Let the angle $\theta$ be the angle formed between lines $AC$ and $BD$.

5. Mathematical Approach:

   • Given $AC$ and $BD$ are doubled from $OA$ and $OB$, respectively, we can write:
     • $AC = 2 \cdot OA$
     • $BD = 2 \cdot OB$
   • To find $\cos \theta$ and $\tan \theta$, we need the direction vectors or coordinate points for the lines.

6. Solution:

   • Assume vectors $\vec{OA} = \vec{a}$, $\vec{OB} = \vec{b}$.
   • $\vec{AC} = \vec{CA} - \vec{OA}$, similarly $\vec{BD} = \vec{DC} - \vec{OB}$.
   • Using vector dot product properties:
     • $\cos \theta = \frac{\vec{AC} \cdot \vec{BD}}{|\vec{AC}| |\vec{BD}|}$
     • Given the nature of the doubling, assume $OA$ and $OB$ are perpendicular for simplicity. Then $\vec{AC} \cdot \vec{BD} = 0$, giving $\cos \theta = 0$.
     • $\tan \theta$ can be derived using geometric relations or trigonometric identities depending on the specific coordinates.

7. Answer Verification:

   • Checking the options provided in the image, it’s crucial to confirm the calculations match one of the choices.
   • The answer provided in the image is option (c) ( \frac