Question

If \( \cos \theta + \sin \theta = x \cos \theta \) and \( \sin \theta = y \cos \theta \), then \( x^2 + y^2 = \)

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. None of these

Hint:

In trigonometric equations, express one variable in terms of others and use algebraic manipulation to solve for unknowns.

Answer 1

The Correct Option is A

Step 1: Express the equations in terms of \( x \) and \( y \).
Given the equations: \[ \cos \theta + \sin \theta = x \cos \theta \quad \text{and} \quad \sin \theta = y \cos \theta \] Substitute \( \sin \theta = y \cos \theta \) into the first equation: \[ \cos \theta + y \cos \theta = x \cos \theta \] Factor out \( \cos \theta \): \[ \cos \theta (1 + y) = x \cos \theta \] Therefore, \( x = 1 + y \).
Step 2: Solve for \( x^2 + y^2 \).
Now, \( x^2 + y^2 = (1 + y)^2 + y^2 = 1 + 2y + y^2 + y^2 = 1 \). Final Answer: \[ \boxed{1} \]