Final answer:
The analogies provided in the question incorrectly mix categories, with 'swim' being an action and 'hockey' being a sport, which makes it hard to draw a parallel. However, extending the analogy pattern correctly, we can conclude that the essential item used in tennis is a 'ball'.
Step-by-step explanation:
The analogy cosk : slope suggests a relationship between a term in trigonometry and its representation in geometry, specifically how the cosine function relates to the slope of a line. Similarly, the analogy swim : hockey :: puck represents the relationship between a sport (hockey) and an essential item used in the sport (a puck). Extending this pattern to tennis, the essential item used in tennis is a 'ball'. Therefore, the complete analogies are: cosk : slope :: hockey : puck :: tennis : ball.
However, the analogies involving swim and the options given (A. gold, B. first, C. bronze, D. thrill) are not structured properly and do not clearly relate to the analogy pattern. The analogies do not form a parallel relationship because 'swim' is an action, whereas 'hockey' is a sport, and the options A through D do not directly correspond to an object in tennis. As such, there is no correct answer from the options given, and the question contains a typo or an irrelevant part.