Step-by-Step Solution
Key Concept: Use the definition of cotangent \(\cot A = \frac{\text{adjacent}}{\text{opposite}}\) and the Pythagorean relation \(\sin^2 A + \cos^2 A = 1\). From \(\cot A\) we obtain \(\tan A\), construct a right‑angled triangle, compute the hypotenuse, and then evaluate \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}}\) and \(\sec A = \frac{\text{hypotenuse}}{\text{adjacent}}\).
1. Given condition
\[15\cot A = 8 \quad\Rightarrow\quad \cot A = \frac{8}{15}.\]
2. Relation between cot and tan
\[\cot A = \frac{1}{\tan A} \;\Rightarrow\; \tan A = \frac{1}{\cot A}=\frac{15}{8}.\]
3. Form a right‑angled triangle
Let the side opposite \(A\) be \(15\) units and the side adjacent to \(A\) be \(8\) units (consistent with \(\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{8}\)).
4. Find the hypotenuse using Pythagoras theorem:
\[\text{hypotenuse} = \sqrt{8^{2}+15^{2}} = \sqrt{64+225}=\sqrt{289}=17.\]
5. Compute \(\sin A\)
\[\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{15}{17}.\]
6. Compute \(\sec A\)
\[\sec A = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{17}{8}.\]
7. Answer
\[\sin A = \frac{15}{17}, \qquad \sec A = \frac{17}{8}.\]
Correct Answer: sin A = 15/17, sec A = 17/8