Question

The value of $\dfrac{1 + \tan^2 A}{1 + \cot^2 A}$ will be:

• A. $\sec^2 A$
• B. -1
• C. $\cot^2 A$
• D. $\tan^2 A$

Hint:

Always replace $1 + \tan^2 A$ with $\sec^2 A$ and $1 + \cot^2 A$ with $\csc^2 A$ for simplification.

Answer 1

The Correct Option is D

Step 1: Use trigonometric identities.
We know $\sec^2 A = 1 + \tan^2 A$ and $\csc^2 A = 1 + \cot^2 A$.

Step 2: Substitute in the expression.
\[ \dfrac{1 + \tan^2 A}{1 + \cot^2 A} = \dfrac{\sec^2 A}{\csc^2 A} = \dfrac{1/\cos^2 A}{1/\sin^2 A} = \dfrac{\sin^2 A}{\cos^2 A} = \tan^2 A \]
Step 3: Conclusion.
Hence, $\dfrac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A$.