Question

\(\frac{1 + \tan^2 A}{1 + \cot^2 A}\) equals to:

• A. \(\tan^2 A\)
• B. \(-1\)
• C. \(-\tan^2 A\)
• D. \(\cot^2 A\)

Hint:

A quick shortcut for such fractions is to remember that \(\cot A = 1/\tan A\). Thus, \(1 + \cot^2 A = 1 + \frac{1}{\tan^2 A} = \frac{\tan^2 A + 1}{\tan^2 A}\). Dividing the numerator by this immediately gives \(\tan^2 A\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires the use of fundamental trigonometric identities.
Step 2: Key Formula or Approach:
Use the identities:
\(1 + \tan^2 A = \sec^2 A\)
\(1 + \cot^2 A = \csc^2 A\)
Step 3: Detailed Explanation:
Substitute the identities into the expression:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \frac{\sec^2 A}{\csc^2 A} \]
Using the definitions of secant and cosecant:
\[ \sec^2 A = \frac{1}{\cos^2 A} \quad \text{and} \quad \csc^2 A = \frac{1}{\sin^2 A} \]
Therefore:
\[ \frac{\sec^2 A}{\csc^2 A} = \frac{1/\cos^2 A}{1/\sin^2 A} = \frac{1}{\cos^2 A} \times \frac{\sin^2 A}{1} = \frac{\sin^2 A}{\cos^2 A} \]
\[ \frac{\sin^2 A}{\cos^2 A} = \tan^2 A \]
Step 4: Final Answer:
The expression equals \(\tan^2 A\).