Question

If $3 \cot A = 4$, then the value of $\dfrac{1 - \tan^2 A}{1 + \tan^2 A}$ will be:
 

• A. $\tfrac{7}{25}$
• B. $-\tfrac{7}{25}$
• C. $\tfrac{8}{17}$
• D. $\tfrac{9}{41}$

Hint:

For trigonometric identities, always check the quadrant of the angle to determine the correct sign of the value.

Answer 1

The Correct Option is B

 

Step 1: Express $\cot A$ 
Given $3 \cot A = 4 $ $\Rightarrow$ $ \cot A = \tfrac{4}{3}$. 
Thus, $\tan A = \tfrac{1}{\cot A} = \tfrac{3}{4}$. 
 

Step 2: Apply the given expression 
We need to evaluate: \[ \dfrac{1 - \tan^2 A}{1 + \tan^2 A} \] Substitute $\tan A = \tfrac{3}{4}$: \[ \tan^2 A = \left(\tfrac{3}{4}\right)^2 = \tfrac{9}{16} \] \[ \dfrac{1 - \tfrac{9}{16}}{1 + \tfrac{9}{16}} = \dfrac{\tfrac{16}{16} - \tfrac{9}{16}}{\tfrac{16}{16} + \tfrac{9}{16}} = \dfrac{\tfrac{7}{16}}{\tfrac{25}{16}} = \dfrac{7}{25} \]

Step 3: Identify trigonometric identity 
We know: \[ \dfrac{1 - \tan^2 A}{1 + \tan^2 A} = \cos 2A \] Since $\tan A = \tfrac{3}{4}$, angle $A$ is acute ($0 < A < \tfrac{\pi}{2}$). 
Thus, $\cos 2A$ will be negative because $2A$ lies in the second quadrant. 
Hence, the correct value is $-\tfrac{7}{25}$. 
 

Step 4: Conclusion 
The required value is $-\tfrac{7}{25}$. 
The correct answer is option (B).