If sinθ / x = cosθ / y, then sinθ − cosθ is:

Question

If sinθ / x = cosθ / y, then sinθ − cosθ is:

1. A. (x − y) / (2√(x² + y²))
2. B. (x − y) / √(3x² + 5y²)
3. C. (5x − y) / (3√(x² + y²))
4. D. (x − y) / √(x² + y²) (Correct answer)

Correct Answer

Option D — (x − y) / √(x² + y²)

Detailed Solution & Explanation

The correct answer is (x − y) / √(x² + y²).

Key Points

• Set the equal ratios to a constant k:
  • sinθ/x = cosθ/y = k, so sinθ = kx and cosθ = ky
• Apply the Pythagorean identity:
  • sin²θ + cos²θ = 1 → k²(x² + y²) = 1 → k = 1/√(x² + y²)
• Therefore sinθ − cosθ = k(x − y) = (x − y) / √(x² + y²).

Additional Information

• Introducing a constant of proportionality is the standard move whenever two ratios are stated equal — it converts a relationship into two usable expressions.
• The identity sin²θ + cos²θ = 1 is what pins k down; without it the problem is underdetermined.
• The same setup gives sinθ + cosθ = (x + y)/√(x² + y²), and multiplying the two results yields (x² − y²)/(x² + y²).
• Note that k could be negative depending on the quadrant, but the options assume the principal positive root, as is conventional in these questions.