Why didn't Light Yagami use the Shinigami eyes? Couldn't it have been an easy task for him? Was it his arrogance and overconfidence?
Alternatively, was he not willing to sacrifice half his lifespan, thinking it would be a loss for him?
Wouldn't it have been a wiser step to choose the Shinigami eyes and make life easier than playing and risking around his life?
From the Death Note Wikia:
Ryuk offers to make a deal with Light for the “Shinigami Eyes.” Light refuses the deal, stating that he needs to live in order to rule over his utopian world. He asks Ryuk if there is anything else that he should know about the Death Note, and Ryuk says probably not.
From watching the anime, we don’t always get insight into Light’s complete train of thought,[1] we only always get to know his conclusion. So those parts of this answer that concern his thoughts are partly based on speculation, depending on what is revealed to the viewer and what is not.
One of Light’s main objectives is to create a utopia[2] in which crime has been eradicated because all criminals are dead. In this utopian world, he — having the power to decide over life and death — would be the unquestioned ruler of all. Pretty much the entire story up to L’s death and Mello and Near’s appearance shows his slow gain of power all as part of his cunning and thoughtful plan.
For most of the early storyline, Light has access to the police’s database through his father[3] allowing him to see both name and face of his victims. While he pretty much instantly acknowledges the power of the Shinigami Eyes, he does get on pretty well without them.
Two points are key points as to why he decides to refuse the Eye deal. The first one is the result of the second paragraph. Light likes being in control, he likes the power. He knows that it will take years to build up his utopia and he wants to enjoy it to the fullest. All of the lifespan he would lose in the deal would be during his rulership. So he’s not just losing half, he is effectively losing more of the ‘good’ time.
The second is the result of the third paragraph. Light is a very bright pupil and underchallenged. He is fascinated by overcoming the obstacles placed in his path. Remember the scene when his room was bugged: he accepted the challenge and continued writing in the death note without the police realising he was doing so. While he could have had his way easily by trading half his lifespan and killing L on first site, it just wouldn’t fit his personality. He would much rather do things the hard and rewarding way.
Note, however, that he does take the easy (or seemingly easy) opportunities if they are presented on a silver plate. When Misa (who he by then has established to be the other death note owner and of whom he knows she traded her lifespan) visits him at the university and gets a glance of L, he immediately tries calling her to ask for his name. He doesn’t even really bother properly leaving L’s earshot to make the call. He probably thought his patience had finally been rewarded and he were now a step closer to the goal. (Ultimately though, this fails because L was already suspicious of Misa and relieved her of her phone.)
Notes:
[1]: I would be surprised if the manga were any different.
[2]: Whether this is utopia or distopia is opinion-based.
[3]: I believe, without his father knowing.
Light wanted to create an utopian world with no crime, and he wanted to Rule the world he created So he needed to live to rule and did not want to shorten his lifespan.
