# Prove |(1,cosx-sinx,cosx+sinx),(1,cosy-siny,cosy+siny),(1,cosz-sinz,cosz+sinz)| =2*|(1,cosx,sinx),(1,cosy,siny),(1,cosz,sinz)|?

Aug 6, 2018

#### Explanation:

Here ,

$L H S = | \left(1 , \cos x - \sin x , \cos x + \sin x\right) , \left(1 , \cos y - \sin y , \cos y + \sin y\right) , \left(1 , \cos z - \sin z , \cos z + \sin z\right) |$

Taking color(red)(C_2+C_3

$L H S = | \left(1 , \cos x - \sin x + \textcolor{red}{\cos x + \sin x} , \cos x + \sin x\right) , \left(1 , \cos y - \sin y + \textcolor{red}{\cos y + \sin y} , \cos y + \sin y\right) , \left(1 , \cos z - \sin z + \textcolor{red}{\cos z + \sin z} , \cos z + \sin z\right) |$

$L H S = | \left(1 , 2 \cos x , \cos x + \sin x\right) , \left(1 , 2 \cos y , \cos y + \sin y\right) , \left(1 , 2 \cos z , \cos z + \sin z\right) |$

Taking color(blue)(C_2(1/2)

$L H S = \textcolor{b l u e}{2} | \left(1 , \textcolor{b l u e}{\cos x} , \cos x + \sin x\right) , \left(1 , \textcolor{b l u e}{\cos y} , \cos y + \sin y\right) , \left(1 , \textcolor{b l u e}{\cos z} , \cos z + \sin z\right) |$

Taking color(violet)(C_3-C_2

$L H S = 2 | \left(1 , \cos x , \cos x + \sin x \textcolor{v i o \le t}{- \cos x}\right) , \left(1 , \cos y , \cos y + \sin y \textcolor{v i o \le t}{- \cos y}\right) , \left(1 , \cos z , \cos z + \sin z \textcolor{v i o \le t}{- \cos z}\right) |$

$\therefore L H S = 2 | \left(1 , \cos x , \sin x\right) , \left(1 , \cos y , \sin y\right) , \left(1 , \cos z , \sin z\right) |$

Hence , $L H S = R H S$