If

sin(x) = 1/3 and sec(y) = 13/12
, where x and y lie between 0 and π/2, evaluate the expression using trigonometric identities. (Enter an exact answer.)
sin(x + y)

To evaluate the expression sin(x + y), we can use the sum identity for sine:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

Given that sin(x) = 1/3 and sec(y) = 13/12, we need to find the values of cos(x) and sin(y) in order to substitute them into the formula.

Let's start with sin(x) = 1/3. Since sin(x) is positive and lies between 0 and π/2, we can use the Pythagorean identity to find cos(x):

cos(x) = sqrt(1 - sin^2(x))
= sqrt(1 - (1/3)^2)
= sqrt(1 - 1/9)
= sqrt(8/9)
= sqrt(8) / sqrt(9)
= 2√2 / 3

Next, we have sec(y) = 13/12. Secant is the reciprocal of cosine, so we can find cos(y) using the reciprocal property:

cos(y) = 1 / sec(y)
= 1 / (13/12)
= 12/13

Finally, we can substitute the values of sin(x), cos(x), cos(y), and sin(y) into the sum identity for sine:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
= (1/3)(12/13) + (2√2 / 3)(sin(y))

And there, we have the expression sin(x + y) evaluated using given trigonometric identities.