In most locality sensitive hashing implemensions of SimHash, why is the cosine distance used and not the euclidean distance?
Cosine distance is common in Information Retrieval and other text-based scenarios because text is most easily represented as high dimensional sparse vectors in the word space. A few specific advantages of cosine distance over Euclidean distance are:


it is fast and simple - particularly, since vectors are sparse, only dimensions present in both vectors need be considered (if the vectors are pre-normalized);
it works well for vectors with moderate overlap in support, whereas in Euclidean distance coordinates outside of the support of one vector dominate;
it tends to work well with a wide variety of techniques - for example, in your question, LSH for cosine distance is simpler than LSH for Euclidean distance;
the magnitude of a vector, which usually corresponds to the length of the document, is not useful - note that cosine distance ignores length but Euclidean distance doesn't;
it is well-studied so it has momentum.