I want to show some function as the norm of two vectors.

Question

I want to see $[\sin(x-y)]^2$ is rewritten by the square of the norm of the difference of two vectors like following (It is a sample and it's not right).
$$ \| (\sin(x) , \cos(x) \sin(x) ) - (\sin(y) , \cos(y) \sin(y) )\| $$ Please tell me how to consider this problem ...

• It is the problem 6.6 of Foundations of Machine Leraning (Mohri etc).

Answer 1

Let $u=\frac{1}{2}(\sin 2x, \cos 2x)$ and $v=\frac{1}{2}(\sin 2y , \cos 2y)$. Then \begin{align*} 2(u-v)&=(\sin 2x-\sin 2y, \cos 2x-\cos 2y)\\ 4\|u-v\|^2&=(\sin 2x-\sin 2y)^2+ (\cos 2x-\cos 2y)^2\\ &=2-2(\sin 2x\sin 2y+\cos 2x\cos 2y)\\ &=2-2\cos(2x-2y)\\ &=2 (1-\cos(2x-2y))\\ &=4 \sin^2(x-y)\\ \|u-v\|^2&=\sin^2(x-y) \end{align*}