Question

Assertion (A): If \((\operatorname{cosec} \theta-\cot \theta)=\frac{1}{3}\) then \((\operatorname{cosec} \theta+\cot \theta)=3\).

Reason (R): \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) for all values of \(\theta\), where \(0^{\circ} \leq \theta \leq 90^{\circ}\).

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. The product of the two expressions is 1 and the first expression is \(\frac{1}{3}\); therefore, the second is 3.

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

Step 3 – Link: Dividing this identity, written as the product of the difference and sum, by the given difference \(\frac{1}{3}\) gives the asserted sum 3. Thus, R explains A.