Question

Assertion (A): If \(\sec 4 \theta=\operatorname{cosec}\left(\theta-10^{\circ}\right)\) and \(4 \theta\) is acute then \(\theta=20^{\circ}\).

Reason (R): \(\sec \phi=\operatorname{cosec}\left(90^{\circ}-\phi\right)\) for all acute angles \(\phi\).

विकल्प

• 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. Applying the identity in R gives \(\operatorname{cosec}(90^{\circ}-4\theta)=\operatorname{cosec}(\theta-10^{\circ})\). Equating the acute angles gives \(90^{\circ}-4\theta=\theta-10^{\circ}\), so \(\theta=20^{\circ}\).

Step 2 – Reason: R is true. It states the complementary-angle identity relating secant and cosecant.

Step 3 – Link: The given equation is converted using R and solving it yields the asserted value. Thus, R correctly explains A.