a) I can't really sketch but I'll try explaining it: 8/17 is a little less than 1/2, and since it's in the fourth quadrant, 2x must be just a little less than 300 degrees.
b) If 2x is a little less than 300, but it's certainly bigger than 270 (indeed cos270=0) so 270<2x<300, which implies 135<x<150, meaning x is in the second quadrant.
c) Here you'd use the identity for cos(2x):
N.B. we only take the negative root because the positive one would give us an angle in the first or 4th quadrants. Check it and see.
d) Just use your calculator to find x using your answer from c).
e) Use your calculator to solve for cosx using the original equation, but don't forget that when you do cosINV(8/17) on your calculator it'll give you the angle in the first quadrant, so you'll need to subtract that angle from 360 (in degrees) or 2pi (in radians) to get the one in the 4th quadrant.
Anyway, hope that helps!