In a right triangle, the sum of the measures of the two acute angles is always 90 degrees. So if ∠J and ∠K are acute angles in a right triangle, then we have:
m∠J + m∠K = 90°
Since m∠J is less than 60°, it follows that m∠K is greater than 30°, because their sum is 90°.
Now, let's consider the cosine of ∠J and ∠K. By definition, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. So we have:
cos(J) = adjacent side to ∠J / hypotenuse
cos(K) = adjacent side to ∠K / hypotenuse
Since ∠J and ∠K are acute angles in a right triangle, they each have a unique adjacent side. Since the triangle is a right triangle, the hypotenuse is always the same, regardless of which angle is being considered.
Therefore, in general, it is not true that cos(J) = cos(K) for all right triangles with acute angles ∠J and ∠K, where m∠J is less than 60°. It depends on the specific values of the adjacent sides.