9 sec2 A – 9 tan2 A =

To solve the expression 9 sec^2 A - 9 tan^2 A, we can use trigonometric identities.

Step 1: Recall that the secant function is the reciprocal of the cosine function, so sec A = 1/cos A. Similarly, the tangent function is the sine function divided by the cosine function, so tan A = sin A / cos A.

Step 2: Substitute these identities into the expression: 9 (1/cos^2 A) - 9 (sin^2 A / cos^2 A).

Step 3: Simplify the expression by finding a common denominator: (9 - 9 sin^2 A) / cos^2 A.

Step 4: Use the Pythagorean identity sin^2 A + cos^2 A = 1 to simplify further: (9 - 9 (1 - cos^2 A)) / cos^2 A.

Step 5: Distribute the negative sign and simplify: (9 - 9 + 9 cos^2 A) / cos^2 A.

Step 6: Combine like terms: (9 cos^2 A) / cos^2 A.

Step 7: Cancel out the common factor of cos^2 A: 9.

Therefore, the expression 9 sec^2 A - 9 tan^2 A simplifies to 9.