Assertion (A): If x = a cos θ + b sin θ and y = a cos θ - b sin θ, then x² + y² = a² + b²

Reason (R): cos²θ + sin²θ = 1.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Given, x = a cos θ + b sin θ and y = a cos θ - b sin θ

⇒ x² = (a cos θ + b sin θ)²

⇒ x² = a² cos² θ + b² sin² θ + 2 ab cos θ sin θ …………………(1)

Similarly,

⇒ y² = (a cos θ - b sin θ)²

⇒ y² = a² cos² θ + b² sin² θ - 2 a b cos θ sin θ …………………(2)

Adding equations (1) and (2), we get :

⇒ x² + y² = (a² cos² θ + b² sin² θ + 2ab cos θ sin θ) + (a² cos² θ + b² sin² θ - 2ab cos θ sin θ)

⇒ x² + y² = (a² cos² θ + a² cos² θ) + (b² sin² θ + b² sin² θ) + (2ab cos θ sin θ - 2ab cos θ sin θ)

⇒ x² + y² = 2a² cos² θ + 2 b² sin² θ

⇒ x² + y² = 2(a² cos² θ + b² sin² θ)

∴ Assertion (A) is false.

cos² θ + sin² θ = 1

This is a fundamental Pythagorean trigonometric identity and is always true for any real value of θ.

∴ Reason (R) is true.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.