Question

If \( x \cos \theta = 1 \) and \( \tan \theta = y \), then the value of \( x^2 - y^2 \) is

• A. 2
• B. 0
• C. -2
• D. 1

Hint:

Use trigonometric identities to simplify and solve complex expressions. Remember, leveraging the Pythagorean identity is crucial in many trigonometric simplifications.

Answer 1

The Correct Option is D

Step 1: From the given, \( x \cos \theta = 1 \) and \( \tan \theta = y \). Step 2: Express \( y \) as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Step 3: Since \( x \cos \theta = 1 \), rewrite \( x \) as \( x = \frac{1}{\cos \theta} \). Step 4: Substitute \( x = \frac{1}{\cos \theta} \) and \( y = \frac{\sin \theta}{\cos \theta} \) into \( x^2 - y^2 \): \[ x^2 - y^2 = \left(\frac{1}{\cos \theta}\right)^2 - \left(\frac{\sin \theta}{\cos \theta}\right)^2 \] Step 5: Simplify the expression: \[ x^2 - y^2 = \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1 - \sin^2 \theta}{\cos^2 \theta} \] Step 6: Use the identity \( 1 - \sin^2 \theta = \cos^2 \theta \): \[ x^2 - y^2 = \frac{\cos^2 \theta}{\cos^2 \theta} = 1 \] Thus, the correct answer is \( \boxed{1} \).