Question

Assertion (A): If \(\operatorname{cosec} \theta=\frac{3}{2}\) then \((\operatorname{cosec}^{2} \theta+\cot ^{2} \theta)=\frac{7}{4}\).

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

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

Explanation:

Step 1 – Assertion: A is false. From R, \(\cot^{2}\theta=\operatorname{cosec}^{2}\theta-1\). Thus, the sum is \(2\operatorname{cosec}^{2}\theta-1=2\left(\frac{3}{2}\right)^{2}-1=\frac{7}{2}\), not \(\frac{7}{4}\).

Step 2 – Reason: R is true; it follows from \(\operatorname{cosec}^{2}\theta-\cot^{2}\theta=1\).