Speaker identification has become a crucial component in various applications, including security systems, virtual assistants, and personalized user experiences. In this paper, we investigate the effectiveness of CosFace Loss and ArcFace Loss for text-independent speaker identification using a Convolutional Neural Network architecture based on the VGG16 model, modified to accommodate mel spectrogram inputs of variable sizes generated from the Voxceleb1 dataset. Our approach involves implementing both loss functions to analyze their effects on model accuracy and robustness, where the Softmax loss function was employed as a comparative baseline. Additionally, we examine how the sizes of mel spectrograms and their varying time lengths influence model performance. The experimental results demonstrate superior identification accuracy compared to traditional Softmax loss methods. Furthermore, we discuss the implications of these findings for future research.