Final Answer:
The reason for this is that the expression cos(xy) can be simplified using trigonometric identities and the given values of sin(x) and sec(y). The correct answer is (c) 3/5.
Step-by-step explanation:
To evaluate cos(xy), we start by expressing sec(y) in terms of cos(y). The identity sec(y) = 1/cos(y) allows us to find cos(y) = 12/13. Now, we use the given value of sin(x) = 1/3. Since sin(x) = 1/csc(x), we find csc(x) = 3. With this information, we can determine cos(x) using the Pythagorean identity sin^2(x) + cos^2(x) = 1. Solving for cos(x), we get cos(x) = 2√2/3.
Now, we want to find cos(xy). Using the product-to-sum trigonometric identity cos(A)cos(B) = 1/2[cos(A+B) + cos(A-B)], we substitute A = x and B = y. This gives us 1/2[cos(x+y) + cos(x-y)]. Since x and y are both between 0 and π/2, we know that x+y and x-y are both within this range.
Now, substituting the values we found earlier, we get 1/2[cos(x+y) + cos(x-y)] = 1/2[cos(x)cos(y) - sin(x)sin(y) + cos(x)cos(y) + sin(x)sin(y)] = cos(x)cos(y). Substituting the values of cos(x) and cos(y), we find cos(xy) = (2√2/3) * (12/13) = 24√2/39.
To simplify the expression, we multiply the numerator and denominator by √2 to rationalize the denominator, resulting in (24√2 * √2) / (39 * √2). This simplifies to 48/78, and further reducing the fraction, we get the final answer: 3/5. The correct answer is (c) 3/5.