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Mathematics

Assertion (A): For 0 < θ ≤ 90°, cosec θ - cot θ and cosec θ + cot θ are reciprocals of each other.

Reason (R): cosec2 θ - cot2 θ = 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 correct, and Reason (R) is the correct reason for Assertion (A).

  4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Trigonometric Identities

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Answer

Solving,

⇒cosec 2θ−cot 2θ⇒1sin 2θ−cos 2θsin 2θ⇒1−cos 2θsin 2θ⇒sin 2θsin 2θ⇒1.\Rightarrow \text{cosec }^2 θ - \text{cot }^2 θ \\[1em] \Rightarrow \dfrac{1}{\text{sin }^2 θ} - \dfrac{\text{cos }^2 θ}{\text{sin }^2 θ} \\[1em] \Rightarrow \dfrac{1 - \text{cos }^2 θ}{\text{sin }^2 θ}\\[1em] \Rightarrow \dfrac{\text{sin }^2 θ}{\text{sin }^2 θ}\\[1em] \Rightarrow 1.

Since, L.H.S. = R.H.S.

So, cosec2 θ - cot2 θ = 1, the condition 0 < θ ≤ 90° ensures that both cosec θ and cot θ are defined and non-zero, making the statement valid.

∴ Reason (R) is true.

⇒ cosec2 θ - cot2 θ = 1

⇒ (cosec θ - cot θ)(cosec θ + cot θ) = 1

⇒ (cosec θ - cot θ) = 1cosec θ + cot θ\dfrac{1}{\text{cosec θ + cot θ}}

∴ Assertion (A) is true.

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

Hence, option 3 is the correct option.

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