To simplify the expression $\frac{\sec^2A - 6\tan A + 7}{\sec^2A} - 5$, follow these steps:

Step 1: Separate the Fraction

The expression can be rewritten as: $\frac{\sec^2A}{\sec^2A} - \frac{6\tan A}{\sec^2A} + \frac{7}{\sec^2A} - 5$

Step 2: Simplify Each Term

• The first term simplifies to $1$ since $\frac{\sec^2A}{\sec^2A} = 1$.
• The second term can be simplified by recognizing that $\sec^2A = 1 + \tan^2A$. Thus, $\frac{6\tan A}{\sec^2A} = \frac{6\tan A}{1 + \tan^2A}$
• The third term simplifies to $\frac{7}{\sec^2A} = 7\cos^2A$ since $\sec^2A = \frac{1}{\cos^2A}$.

Step 3: Combine and Simplify the Expression

So, the expression becomes: $1 - \frac{6\tan A}{1 + \tan^2A} + 7\cos^2A - 5$

Finally, simplify further: $= (1 - 5) + 7\cos^2A - \frac{6\tan A}{1 + \tan^2A}$ $= -4 + 7\cos^2A - \frac{6\tan A}{1 + \tan^2A}$

Final Simplified Expression:

$\boxed{-4 + 7\cos^2A - \frac{6\tan A}{1 + \tan^2A}}$

Would you like more details on any part of this simplification? Here are some related questions:

1. How do you simplify trigonometric identities?
2. What is the relationship between $\sec^2A$ and $\tan^2A$?
3. Can you explain how to simplify expressions involving $\sec A$ and $\tan A$?
4. How would this expression change if you were given a specific value for $A$?
5. What are some common tricks to simplifying complex trigonometric expressions?

Tip: Always check if a trigonometric identity can simplify the expression before performing algebraic operations.