Given:
Cos^2A - sin^2A = tan^2B

To prove:
Cos^2B - sin^2B = tan^2A

Proof:

Step 1: Rewrite the given equation using trigonometric identities
Using the Pythagorean identity, sin^2A + cos^2A = 1, we can rewrite the given equation as:
cos^2A - (1 - cos^2A) = tan^2B
Simplifying, we get:
2cos^2A - 1 = tan^2B

Step 2: Express tan^2B in terms of tanA
Using the identity tan^2θ = sec^2θ - 1, we can express tan^2B in terms of cosB:
tan^2B = sec^2B - 1

Step 3: Substitute tan^2B in the equation
Substituting the value of tan^2B from Step 2 into the equation in Step 1, we have:
2cos^2A - 1 = sec^2B - 1

Step 4: Simplify the equation
Cancelling out the common terms on both sides of the equation, we get:
2cos^2A = sec^2B

Step 5: Express sec^2B in terms of cosB
Using the identity sec^2θ = 1 + tan^2θ, we can express sec^2B in terms of cosB:
sec^2B = 1 + tan^2B

Step 6: Substitute sec^2B in the equation
Substituting the value of sec^2B from Step 5 into the equation in Step 4, we have:
2cos^2A = 1 + tan^2B

Step 7: Express tan^2B in terms of tanA
Using the identity tan^2θ = sec^2θ - 1, we can express tan^2B in terms of cosB:
1 + tan^2A = 1 + tan^2B

Step 8: Simplify the equation
Cancelling out the common terms on both sides of the equation, we get:
tan^2A = tan^2B

Step 9: Rearrange the equation
Rearranging the equation, we have:
tan^2B = tan^2A

Step 10: Express cos^2B - sin^2B in terms of tan^2A
Using the identity cos^2θ - sin^2θ = 1 - tan^2θ, we can express cos^2B - sin^2B in terms of tan^2A:
cos^2B - sin^2B = 1 - tan^2A

Conclusion:
Therefore, we have proved that cos^2B - sin^2B = tan^2A.