Question

Assertion (A): If \(\cos 2 \theta=\sin \left(\theta-15^{\circ}\right)\) then \(\theta=35^{\circ}\).

Reason (R): \(\cos \theta=\sin \left(90^{\circ}-\theta\right)\), for \(0^{\circ} \leq \theta \leq 90^{\circ}\).

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. Replacing cosine with the sine of the complementary angle gives \(\sin(90^{\circ}-2\theta)=\sin(\theta-15^{\circ})\). Equating the angles yields \(90^{\circ}-2\theta=\theta-15^{\circ}\), so \(\theta=35^{\circ}\).

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

Step 3 – Link: The equation is transformed using the identity stated in R, which leads to the asserted value; hence, R correctly explains A.