To expand \(\sec(\pi + A)\), we'll first use the identity for the secant function and then use a trigonometric identity for angles:

1. **Secant Identity**:
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)}
   \]

2. **Cosine Angle Addition Identity**:
   \[
   \cos(\pi + A) = -\cos(A)
   \]
   
   This is because the cosine of an angle plus \(\pi\) (180 degrees) is the negative of the cosine of the original angle.

Using these identities:

\[
\sec(\pi + A) = \frac{1}{\cos(\pi + A)} = \frac{1}{-\cos(A)} = -\sec(A)
\]

So, the expanded form is:

\[
\sec(\pi + A) = -\sec(A)
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. What is the expanded form of \(\sin(\pi + A)\)?
2. How would you simplify \(\sec(\pi/2 + A)\)?
3. What is the general rule for trigonometric functions when adding \(\pi\) to an angle?
4. Can you express \(\csc(\pi + A)\) in terms of \(\csc(A)\)?
5. How would you expand \(\sec(2\pi + A)\)?

**Tip:** Understanding how trigonometric functions transform with different angle shifts (like adding \(\pi\), \(\pi/2\), etc.) is crucial for solving more complex trigonometric problems.

Explore how to expand secant function sec(π + A) using trigonometric identities and cosine angle addition. Understand the fundamental principles of trigonometry and apply them to solve complex trigonometric expressions. Suitable for advanced high school students and those studying trigonometry.

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