LLM-powered conversational agents are increasingly influencing our decision-making, raising concerns about "sycophancy" - the tendency for LLMs to excessively agree with users even at the expense of truthfulness. While prior work has primarily examined LLM sycophancy as a model behavior, our understanding of how users perceive this phenomenon and its impact on user trust remains significantly lacking. In this work, we conceptualize LLM sycophancy along two key constructs: conversational demeanor (complimentary vs. neutral) and stance adaptation (adaptive vs. consistent). A 2 x 2 between-subjects experiment (N = 224) revealed complex dynamics: complimentary LLMs that adapted their stance reduced perceived authenticity and trust, while neutral LLMs that adapted enhanced both, suggesting a pathway for manipulating users into over-trusting LLMs beyond their actual capabilities. Our findings advance user-centric understanding of LLM sycophancy and provide profound implications for developing more ethical and trustworthy LLM systems.