Question

\(\frac{1 + \tan^2 A}{1 + \cot^2 A} = \)

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

Hint:

A useful shortcut to remember is that \(\frac{\sec^2 A}{\csc^2 A} = \tan^2 A\). This can save a few steps in the simplification process.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves simplifying a trigonometric expression using the Pythagorean identities.

Step 2: Key Formula or Approach:
We will use the two Pythagorean identities:
1. \(1 + \tan^2 A = \sec^2 A\)
2. \(1 + \cot^2 A = \csc^2 A\)
And the reciprocal identities \(\sec A = 1/\cos A\) and \(\csc A = 1/\sin A\).

Step 3: Detailed Explanation:
Start with the given expression:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} \] Substitute the Pythagorean identities in the numerator and the denominator:
\[ = \frac{\sec^2 A}{\csc^2 A} \] Now, use the reciprocal identities to express secant and cosecant in terms of sine and cosine:
\[ = \frac{1/\cos^2 A}{1/\sin^2 A} \] To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator:
\[ = \frac{1}{\cos^2 A} \times \frac{\sin^2 A}{1} = \frac{\sin^2 A}{\cos^2 A} \] Using the identity \(\tan A = \sin A / \cos A\), we get:
\[ = \tan^2 A \]

Step 4: Final Answer:
The value of the expression is \(\tan^2 A\).