Question

(1+cosx)(1+cosy)(1+cosz)-(1-cosx)(1-cosy)(1-cosz)=k then k= ?

Answer 1

I know it's a pain, but if you (egads) multiply out the LHS, then you'll see that there's a LOT of cancellation.

Answer 2

For example:
(1+cosx)(1+cosy)(1+cosz)
= (1 + cosx + cosy + cosx cosy)(1 + cosz)
= 1 + cosz + cosx + cosx cos z + cosy + cosy cosz + cosx cosy + cosx cosy cosz
Do the same with second term; and watch things disappear!

Answer 3

well, (1+cosx)(1+cosy)(1+cosz)-(1-cosx)(1-cosy)(1-cosz)
= (1+ cosy + cosx + cosx cosy) (1+cosz) - [ (1- cosy - cosx + cosx cosy) (1-cosz)]
= (1+cosy+cosx+cosx cosy+cosz+cosycosz+cosxcosz+cosxcosycosz) - [1-cosy-cosx+cosxcosy-cosz+cosycosz+cosxcosz-cosxcosycosz]
= 2cosy+2cosx+2cosz+2cosxcosycosz
factoring out 2:
= 2 (cosy+cosx+cosz+cosxcosycosz)
therefore: k = 2 (cosy+cosx+cosz+cosxcosycosz)