The sum of the legs of a right triangle is 17 an hypothenuse is 13. What is the product of cosines of acute angles?

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.