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Mathematics

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).

Trigonometrical Ratios

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Answer

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.

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