`sec (pi + theta)` is -

Options

A. `cos theta`
B. `tan theta`
C. `sec theta`
D. `-sec theta`

Answer

To solve the problem of finding \( \sec(\pi + \theta) \), we can follow these steps:

### Step 1: Understand the definition of secant
The secant function is defined as:
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \]
Thus, we can express \( \sec(\pi + \theta) \) as:
\[ \sec(\pi + \theta) = \frac{1}{\cos(\pi + \theta)} \]

### Step 2: Use the cosine addition formula
We need to find \( \cos(\pi + \theta) \). According to the cosine addition formula, we have:
\[ \cos(\pi + \theta) = -\cos(\theta) \]
This is a standard trigonometric identity.

### Step 3: Substitute the cosine value into the secant expression
Now we can substitute \( \cos(\pi + \theta) \) into our expression for secant:
\[ \sec(\pi + \theta) = \frac{1}{\cos(\pi + \theta)} = \frac{1}{-\cos(\theta)} \]

### Step 4: Simplify the expression
This simplifies to:
\[ \sec(\pi + \theta) = -\frac{1}{\cos(\theta)} \]
We know that \( \frac{1}{\cos(\theta)} \) is equal to \( \sec(\theta) \):
\[ \sec(\pi + \theta) = -\sec(\theta) \]

### Conclusion
Thus, the final result is:
\[ \sec(\pi + \theta) = -\sec(\theta) \]

### Answer
The correct answer is \( -\sec(\theta) \).

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