# Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$

$$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 \sin(\theta) + v_2 \cos(\theta))$$

My question is what was the simplification step between the two lines? I can't seem to simplify the magnitude of the vector $V$ to get that final expression.

I suppose $v_1$ and $v_2$ are the rectangular coordinates of vector $V$, and $\varphi$ is its angle in polar coordinates.
Thus $v_1=||V||\cos\varphi$ and $v_2=||V||\sin\varphi$.