Question

Assertion (A): If \(\sin \alpha=\frac{1}{2}\) then \((3 \cos \alpha - 4 \cos^{3} \alpha) = 0\).

Reason (R): \(\cos 3 \alpha = 3 \cos \alpha - 4 \cos^{3} \alpha\).

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. Since \(\sin\alpha=\frac{1}{2}\), the acute-angle value is \(\alpha=30^{\circ}\), so \(\cos 3\alpha=\cos 90^{\circ}=0\). By the identity in R, \(3\cos\alpha-4\cos^{3}\alpha=0\).

Step 2 – Reason: R is true. It is the triple-angle identity for cosine.

Step 3 – Link: Substituting the given angle condition into the identity in R proves the expression in A is zero.