Question

Assertion (A): The value of \((\cot 10^{\circ} \times \cot 30^{\circ} \times \cot 80^{\circ})\) is \(\frac{\sqrt{3}}{2}\).

Reason (R): \(\cot \left(90^{\circ}-\theta\right)=\tan \theta\).

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. Since \(80^{\circ}=90^{\circ}-10^{\circ}\), the product \(\cot 10^{\circ}\times\cot 80^{\circ}=1\). Also, \(\cot 30^{\circ}=\sqrt{3}\). Hence, the full product is \(\sqrt{3}\), not \(\frac{\sqrt{3}}{2}\).

Step 2 – Reason: R is true; it is the complementary-angle identity for cotangent and tangent.