For all `theta, tan theta+ cos theta + tan (-theta) + cos(-theta)=`

Options

A. 0
B. `2 tan theta`
C. `2 cos theta `
D. `2(tan theta + cos theta)`

Explanation

To solve the expression \( \tan \theta + \cos \theta + \tan(-\theta) + \cos(-\theta) \), we can follow these steps:

### Step 1: Identify the properties of trigonometric functions
We know that:
- \( \tan(-\theta) = -\tan(\theta) \) (odd function)
- \( \cos(-\theta) = \cos(\theta) \) (even function)

### Step 2: Substitute the properties into the expression
Using these properties, we can rewrite the expression:
\[ \tan \theta + \cos \theta + \tan(-\theta) + \cos(-\theta) = \tan \theta + \cos \theta + (-\tan \theta) + \cos \theta \]

### Step 3: Simplify the expression
Now, let's simplify the expression:
\[ \tan \theta - \tan \theta + \cos \theta + \cos \theta = 0 + 2\cos \theta = 2\cos \theta \]

### Final Answer
Thus, the expression simplifies to:
\[ \tan \theta + \cos \theta + \tan(-\theta) + \cos(-\theta) = 2\cos \theta \]