Your question answers itself.
You have shown the correct answer for the first drop-down: -161/289
You have shown the 3 incorrect answers out of the 4 choices for the second drop-down. The correct choice is the remaining one: -240/161.
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Even if all you know is that tan = sin/cos, you would suspect that answer choice based on the numerator of the cosine: 161. That value is likely the one in the denominator of the tangent value.
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cos(2θ) = 2cos(θ)² . . . . trig identity
... = 2(-8/17)² -1 = 128/289 -1 = -161/289
sin(2θ) = √(1-cos(2θ)²) . . . . trig identity
... = √(289² -161²)/289 = √57600/289 = 240/289
tan(2θ) = sin(2θ)/cos(2θ) . . . . trig identity
... = (240/289)/(-161/289) = -240/161
