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

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Answer 1 · 2018-08-06T19:51:45

Please see below.

Here ,

$LHS=|(1,cosx-sinx,cosx+sinx),(1,cosy-siny,cosy+siny),(1,cosz-sinz,cosz+sinz)|$

Taking $color(red)(C_2+C_3$

$LHS=|(1,cosx-sinx+color(red)(cosx+sinx),cosx+sinx),(1,cosy-siny+color(red)(cosy+siny),cosy+siny),(1,cosz-sinz+color(red)(cosz+sinz),cosz+sinz)|$

$LHS=|(1,2cosx,cosx+sinx),(1,2cosy,cosy+siny),(1,2cosz,cosz+sinz)|$

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

$LHS=color(blue)(2)|(1,color(blue)(cosx),cosx+sinx),(1,color(blue)(cosy),cosy+siny),(1,color(blue)(cosz),cosz+sinz)|$

Taking $color(violet)(C_3-C_2$

$LHS=2|(1,cosx,cosx+sinxcolor(violet)(-cosx)),(1,cosy,cosy+sinycolor(violet)(-cosy)),(1,cosz,cosz+sinzcolor(violet)(-cosz))|$

$:.LHS=2|(1,cosx,sinx),(1,cosy,siny),(1,cosz,sinz)|$

Hence , $LHS=RHS$

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)||(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)|?