If sec(∞) < 0 and the sin(∞) = 15/17 what is the exact value of cot (∞)

The numbers are 9 and 10.

Let x be the first number.  Since they are consecutive, this means (x+1) represents the second number.

The square of x is x².  The square of (x+1) is
(x+1)²=(x+1)(x+1)=x²+1x+1x+1=x²+2x+1

The difference of these squares would be given by
x²+2x+1 - x²

Setting this equal to 19,
x²+2x+1 - x² = 19

Combining like terms,
2x+1=19

Subtract 1 from each side:
2x+1-1=19-1
2x=18

Divide both sides by 2:
2x/2 = 18/2
x = 9

The first number, x, is 9.  This means the second number, x+1, is 9+1=10.