Please indicate the source:
https://www.cnblogs.com/darkknightzh/p/12013741.html
Online references a lot, and summarize it.
Two vectors $ \ mathbf {A} = [{{a} _ {1}}, \ cdots, {{a} _ {n}}] $, $ \ mathbf {B} = [{{b} _ {1 }}, \ cdots, {{b} _ {n}}] $, Euclidean distance between two vectors:
$Euc\_dist={{\left\| \mathbf{A}-\mathbf{B} \right\|}_{2}}=\sqrt{\sum\limits_{i=1}^{n}{{{({{a}_{i}}-{{b}_{i}})}^{2}}}}=\sqrt{\sum\limits_{i=1}^{n}{(a_{i}^{2}-2\centerdot {{a}_{i}}\centerdot {{b}_{i}}+b_{i}^{2})}}=\sqrt{\sum\limits_{i=1}^{n}{a_{i}^{2}}+\sum\limits_{i=1}^{n}{b_{i}^{2}}-2\centerdot \sum\limits_{i=1}^{n}{{{a}_{i}}\centerdot {{b}_{i}}}}$
The two cosine similarity between vectors Cos_sim as:
$Cos\_sim\text{=}\frac{\mathbf{A}\centerdot {{\mathbf{B}}^{T}}}{{{\left\| \mathbf{A} \right\|}_{2}}\centerdot {{\left\| \mathbf{B} \right\|}_{2}}}=\frac{\sum\limits_{i=1}^{n}{{{a}_{i}}\centerdot {{b}_{i}}}}{\sqrt{\sum\limits_{i=1}^{n}{a_{i}^{2}}}\centerdot \sqrt{\sum\limits_{i=1}^{n}{b_{i}^{2}}}}$
Cosine distance:
$Cos\_dis\text{=}1-Cos\_sim\text{=}1\text{-}\frac{\sum\limits_{i=1}^{n}{{{a}_{i}}\centerdot {{b}_{i}}}}{\sqrt{\sum\limits_{i=1}^{n}{a_{i}^{2}}}\centerdot \sqrt{\sum\limits_{i=1}^{n}{b_{i}^{2}}}}$
If the two vectors have been normalized, i.e., $ {{\ left \ | \ mathbf {A} \ right \ |} _ {2}} = \ sqrt {\ sum \ limits_ {i = 1} ^ {n } {a_ {i} ^ {2}}} \ text {=} 1 $, $ {{\ left \ | \ mathbf {B} \ right \ |} _ {2}} = \ sqrt {\ sum \ limits_ {i = 1} ^ {n} {b_ {i} ^ {2}}} \ text {=} 1 $, then:
$Euc\_dist\text{=}\sqrt{1+1-2\centerdot \sum\limits_{i=1}^{n}{{{a}_{i}}\centerdot {{b}_{i}}}}\text{=}\sqrt{2\centerdot 1-\sum\limits_{i=1}^{n}{{{a}_{i}}\centerdot {{b}_{i}}}}$
$Cos\_dis\text{=}1\text{-}\sum\limits_{i=1}^{n}{{{a}_{i}}\centerdot {{b}_{i}}}$
and then:
$Euc\_dis{{t}^{2}}\text{=}2\centerdot Cos\_dis\text{=}2\centerdot (1-Cos\_sim)$
$Cos\_sim=1-\frac{1}{2}Euc\_dis{{t}^{2}}$