If 3cotA = 4, check whether (1−tan²A)/(1+tan²A) = cos2A − sin2A or not.

Let B = 90°
cotA = AB/BC = 4/3
Let AB = 4x, BC = 3x.
AC² = AB² + BC²
AC² = 16x² + 9x²
AC² = 25x²
AC = 5x
Now, tanA = 3/4
LHS : (1−tan²A)/(1+tan²A) = (1 − 9/16)/(1 + 9/16) = 7/25
RHS : cos2A − sin2A = (4/5)² − (3/5)² = 16/25 − 9/25 = 7/25
⇒ LHS = RHS
∴ (1−tan²A)/(1+tan²A) = cos2A − sin2A