Let b and c be the legs of the right angle triangle and a is the hypothenuse.
According to enunciation, the sum of the legs is:
b + c = 17
The hypotenuse is of 13 units: a = 13
Let B and C be the acute angles of the right triangle.
cos B = adjacent leg/hypotenuse = c/a
cos C = b/a
We'll multiply cos B by cos C:
cos B*cos C = c*b/a^2
We'll apply Pythagorean theorem in a right angle triangle:
a^2 = b^2 + c^2
But b^2 + c^2 = (b+c)^2 - 2bc
a^2 = (b+c)^2 - 2bc
13^2 = 17^2 - 2bc
We'll isolate 2bc to the left:
2bc = 17^2 - 13^2
2bc = (17 - 13)(17 + 13)
bc = 4*30/2
bc = 60
cos B*cos C = bc/a^2 = 60/169
The requested value of the product cos B*cos C is: cos B*cos C = 60/169.