Question

Assertion (A): If \(\sec \theta=\frac{25}{7}\) then \(\operatorname{cosec} \theta=\frac{25}{24}\).

Reason (R): In a \(\triangle A B C\), right angled at B, we have \(\sec \theta=\frac{A C}{A B}\) and \(\operatorname{cosec} \theta=\frac{A C}{B C}\). 

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. Let the hypotenuse and adjacent side be 25k and 7k. The remaining side is \(\sqrt{(25k)^2-(7k)^2}=24k\). Therefore, \(\operatorname{cosec}\theta=\frac{25k}{24k}=\frac{25}{24}\).

Step 2 – Reason: R is true. The stated ratios define secant and cosecant in the right triangle.

Step 3 – Link: Using the side ratios in R and the Pythagorean theorem gives the asserted cosecant value, so R correctly explains A.