Let's simplify the expression $\frac{\sec^2A - 6\tan A + 7}{\sec^2A - 5}$.

Step 1: Identify Trigonometric Identities

Recall that $\sec^2A = 1 + \tan^2A$.

Step 2: Substitute $\sec^2A = 1 + \tan^2A$

Substitute $\sec^2A = 1 + \tan^2A$ into the expression:

$\frac{(1 + \tan^2A) - 6\tan A + 7}{(1 + \tan^2A) - 5}$

Step 3: Simplify the Numerator and Denominator

Simplify both the numerator and the denominator:

• Numerator: $1 + \tan^2A - 6\tan A + 7 = \tan^2A - 6\tan A + 8$
• Denominator: $1 + \tan^2A - 5 = \tan^2A - 4$

Now the expression is: $\frac{\tan^2A - 6\tan A + 8}{\tan^2A - 4}$

Step 4: Factor the Numerator and Denominator (if possible)

Let's try factoring both the numerator and the denominator:

• Numerator: $\tan^2A - 6\tan A + 8$ can be factored if we find two numbers that multiply to $8$ and add to $-6$. However, no such factors exist for simple factoring.

• Denominator: $\tan^2A - 4$ is a difference of squares, so it can be factored as: $(\tan A - 2)(\tan A + 2)$

So the expression becomes: $\frac{\tan^2A - 6\tan A + 8}{(\tan A - 2)(\tan A + 2)}$

Step 5: Check for Further Simplification

The numerator $\tan^2A - 6\tan A + 8$ does not factor neatly into terms involving $\tan A - 2$ or $\tan A + 2$, so further simplification by canceling terms is not possible.

Final Simplified Expression:

Thus, the final simplified form of the expression is: $\boxed{\frac{\tan^2A - 6\tan A + 8}{(\tan A - 2)(\tan A + 2)}}$

Would you like to explore any part of this simplification in more detail? Here are some related questions:

1. How do you factor quadratic expressions in trigonometry?
2. What are some special factoring techniques for trigonometric functions?
3. Can the quadratic formula be applied to trigonometric expressions?
4. How would this expression behave for specific values of $A$?
5. What role do trigonometric identities play in simplifying complex expressions?

Tip: Always consider whether the expression can be factored or if it can be expressed in a simpler trigonometric identity to aid in simplification.