If \(3 \ cot \ A = 4\), check whether \(\frac{1 - tan^2 \ A}{1 + tan^2 \ A} = cos^2 \ A - sin^2 \ A\) or not.
Proof:
Given \(3 \ cot \ A = 4\).
\(cot \ A=\)
Let us consider the right-angled triangle \(ABC\).
If \(AB = 4k\) and \(BC = 3k\), where \(k\) is a positive number.
Then, \(AC=\)
\(sin \ A =\)
\(cos \ A =\)
\(tan \ A =\)
Consider LHS.
\(\frac{1 - tan^2 \ A}{1 + tan^2 \ A} =\)
Now, let us consider RHS.
\(cos^2 \ A - sin^2 \ A = \left(\frac{4}{5} \right)^2 - \left(\frac{3}{5} \right)^2\)
Therefore, LHS \(=\) RHS.
Hence, we proved.