Question

If \( \cot^{-1} \left( \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \right) \), then find \( \sec^2 \theta \).

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

Use trigonometric identities like \( \cot^2 \theta + 1 = \csc^2 \theta \) and \( \sec^2 \theta = 1 + \tan^2 \theta \) to relate functions and solve for the desired values.

Answer 1

The Correct Option is B

We are given that: \[ \cot^{-1} \left( \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \right) = \theta \] We need to find \( \sec^2 \theta \).

1. Step 1: Express \( \cot \theta \) in terms of \( x \). By the definition of inverse cotangent: \[ \cot \theta = \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \]

2. Step 2: Use the Pythagorean identity. We know that: \[ \cot^2 \theta + 1 = \csc^2 \theta \] Using the identity for \( \csc^2 \theta \): \[ \csc^2 \theta = \left( \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \right)^2 + 1 \] Simplifying: \[ \csc^2 \theta = \frac{1 - x}{1 + x} + 1 = \frac{1 - x + 1 + x}{1 + x} = \frac{2}{1 + x} \]

3. Step 3: Find \( \sec^2 \theta \). We know that: \[ \sec^2 \theta = 1 + \tan^2 \theta \] Using the identity for \( \sec^2 \theta \), we find that \( \sec^2 \theta = 2 \). Thus, the value of \( \sec^2 \theta \) is \( 2 \).