Question

If \( A = 30^\circ \), then the value of \( \frac{1 + \tan^2 A}{1 + \cot^2 A} \) will be:

• A. \( \frac{4}{3} \)
• B. \( -1 \)
• C. 3
• D. \( \frac{1}{3} \)

Hint:

When working with trigonometric identities, use known values for angles like \( 30^\circ \) to simplify the expressions.

Answer 1

The Correct Option is D

Step 1: Use the trigonometric identities.
We are given \( A = 30^\circ \). The identity for \( \tan^2 A \) and \( \cot^2 A \) is: \[ \tan^2 A = \left( \frac{\sin A}{\cos A} \right)^2 \quad \text{and} \quad \cot^2 A = \left( \frac{\cos A}{\sin A} \right)^2. \] But we can simplify this question using known values of trigonometric functions for \( 30^\circ \): \[ \tan 30^\circ = \frac{1}{\sqrt{3}} \quad \text{and} \quad \cot 30^\circ = \sqrt{3}. \]
Step 2: Substitute the values into the expression.
Now, substitute \( \tan^2 30^\circ = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3} \) and \( \cot^2 30^\circ = (\sqrt{3})^2 = 3 \) into the expression: \[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \frac{1 + \frac{1}{3}}{1 + 3} = \frac{\frac{4}{3}}{4} = \frac{4}{3} \times \frac{1}{4} = \frac{1}{3}. \]
Step 3: Conclusion.
Therefore, the value of \( \frac{1 + \tan^2 A}{1 + \cot^2 A} \) is \( \frac{1}{3} \).