The widespread proliferation of online content has intensified concerns about clickbait, deceptive or exaggerated headlines designed to attract attention. While Large Language Models (LLMs) offer a promising avenue for addressing this issue, their effectiveness is often hindered by Sycophancy, a tendency to produce reasoning that matches users' beliefs over truthful ones, which deviates from instruction-following principles. Rather than treating sycophancy as a flaw to be eliminated, this work proposes a novel approach that initially harnesses this behavior to generate contrastive reasoning from opposing perspectives. Specifically, we design a Self-renewal Opposing-stance Reasoning Generation (SORG) framework that prompts LLMs to produce high-quality agree and disagree reasoning pairs for a given news title without requiring ground-truth labels. To utilize the generated reasoning, we develop a local Opposing Reasoning-based Clickbait Detection (ORCD) model that integrates three BERT encoders to represent the title and its associated reasoning. The model leverages contrastive learning, guided by soft labels derived from LLM-generated credibility scores, to enhance detection robustness. Experimental evaluations on three benchmark datasets demonstrate that our method consistently outperforms LLM prompting, fine-tuned smaller language models, and state-of-the-art clickbait detection baselines.