Question

Assertion (A): For \(0<\theta \leq 90^{\circ},(\operatorname{cosec} \theta-\cot \theta)\) and \((\operatorname{cosec} \theta+\cot \theta)\) are reciprocal of each other.

Reason (R): \((\operatorname{cosec}^{2} \theta-\cot ^{2} \theta)=1\).

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (А).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. Two quantities are reciprocals when their product is 1.

Step 2 – Reason: R is true. It gives the identity \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\).

Step 3 – Link: \((\operatorname{cosec} \theta-\cot \theta\)\(\operatorname{cosec} \theta+\cot \theta)=\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\). Thus, R correctly explains A.