#csc(x)# is defined as the reciprocal of #sin(x)#, i.e.:
#csc(x) = 1/sin(x)#
So if #csc(theta) = 17/8#,
That means #1/sin(theta) = 17/8#.
Simplifying a little shows that #sin(theta) = 8/17#
If you now construct a triangle with the given information that #sin(theta) = 8/17#, You get a triangle with a leg length 8, and hypotenuse 17. Using the Pythagorean Theorem, you can solve for the last leg's length:
#a^2 + b^2 = c^2#
#a^2 + 8^2 = 17^2#
#a^2 + 64 = 289#
#a^2 = 225#
#a = 15#
Now you have a triangle, sidelengths 8, 15, and 17, where #theta# adjacent to the 15 side and opposite the 8 side (and the hypotenuse is 17). #cos# means adjacent over hypotenuse, and we just said that the adjacent side is 15 and the hypotenuse is 17, so:
#cos(theta) = 15/17#
