Using your knowledge of trigonometric identities, find the exact value of sin(x+y) knowing that x and y are between 0 and π2, that sin(x)=725 and that sec(y)=1312. Your answer must be a rational number, written in the form of an irreducible fraction p/q (such as 2/3, 4/5, −1/4, etc.), with q>0. Give the integers p and q.

Begin with:

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

To find the exact value of sin(x + y) given that sin(x) = 7/25 and sec(y) = 13/12, we can start by using the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find the value of cos(x).

Since sin(x) = 7/25, we can square both sides of the equation to find sin^2(x):
sin^2(x) = (7/25)^2 = 49/625

Next, we can subtract sin^2(x) from 1 to find cos^2(x):
cos^2(x) = 1 - sin^2(x)
cos^2(x) = 1 - 49/625
cos^2(x) = (625 - 49)/625
cos^2(x) = 576/625

Taking the square root of both sides gives us the value of cos(x):
cos(x) = ±√(576/625)
Since x is between 0 and π/2, cos(x) is positive.

cos(x) = √(576/625)
cos(x) = 24/25

Now, we can use the given value of sec(y) = 13/12 to find the value of cos(y). Recall that sec(y) = 1/cos(y), so we can write:

1/cos(y) = 13/12
Cross-multiplying:

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

Since cos(y) is positive, y must be between 0 and π/2.

Now, we have the values of sin(x) = 7/25, cos(x) = 24/25, and cos(y) = 12/13. To find sin(x + y), we can use the identity:

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

Substituting the given values:

sin(x + y) = (7/25)(12/13) + (24/25)(√(1 - (12/13)^2))

Simplifying further:

sin(x + y) = (84/325) + (24/25)(√(1 - 144/169))
sin(x + y) = (84/325) + (24/25)(√(25/169))

Looking at the term (√(25/169)), we can simplify it further:

√(25/169) = 5/13

Substituting back into the equation:

sin(x + y) = (84/325) + (24/25)(5/13)
sin(x + y) = (84/325) + (24/65)
sin(x + y) = (84/325) + (48/325)
sin(x + y) = 132/325

Therefore, the exact value of sin(x + y) is 132/325, where p = 132 and q = 325.