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.

Learn how to simplify the trigonometric expression (sec^2A - 6tanA + 7) / sec^2A - 5 with detailed steps and explanations. This problem involves applying trigonometric identities and simplification techniques. Suitable for high school students in Grades 10-12.

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