If 3 cot A = 4, check whether 1- tan² A / 1+ tan² A= cos²A – sin² A or not.

Given that: 3 cot A = 4
⇒ cot A = 4/3
Let cot A = 4k/3k, where k is a real number.
In △ABC, by Pythagoras theorem, we have
AC² = BC²+AB²
= (3k)²+(4k)²
= 9k²+16k²
= 25k²
⇒ AC = √(25k²) = 5k
Therefore,
(1-tan²A)/(1+tan²A) = (1-(3/4)²)/(1+(3/4)²) = (1-(9/16))/(1+(9/16)) = ((16-9)/16)/((16+9)/16) = 7/25
and
cos² A-sin²A = (4/5)²-(3/5)² = 16/25 – 9/25 = (16-9)/25 = 7/25
Hence, (1-tan²A)/(1+tan²A) = cos²A-sin²A

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