Given:
3 cot A = 4

To prove:
(1 - tan^2A) / (1 + tan^2A) = cos^2A - sin^2A

Proof:

Step 1: Expressing cot A in terms of sin A and cos A
Recall that cot A is the reciprocal of tan A. We know that tan A = sin A / cos A. Therefore, cot A = 1 / tan A = cos A / sin A.

Given: 3 cot A = 4
Therefore, 3(cos A / sin A) = 4
Multiplying both sides by sin A, we get:
3 cos A = 4 sin A

Step 2: Expressing tan A in terms of sin A and cos A
Recall that tan A = sin A / cos A. Therefore, tan^2A = (sin A / cos A)^2 = sin^2A / cos^2A.

Step 3: Simplifying the expression (1 - tan^2A) / (1 + tan^2A)
Substituting the value of tan^2A from Step 2:
(1 - sin^2A / cos^2A) / (1 + sin^2A / cos^2A)
(1 - sin^2A) / (cos^2A) / (1 + sin^2A) / (cos^2A)
[(1 - sin^2A) / (cos^2A)] * [(cos^2A) / (1 + sin^2A)]
(1 - sin^2A) / (1 + sin^2A)

Step 4: Simplifying the expression cos^2A - sin^2A
Using the trigonometric identity: cos^2A - sin^2A = cos(2A)
cos^2A - sin^2A = cos(2A)

Step 5: Comparing the expressions
(1 - sin^2A) / (1 + sin^2A) = cos(2A)

Conclusion:
We have proved that (1 - tan^2A) / (1 + tan^2A) = cos^2A - sin^2A.