Let `x ,\ y ,\ z in [-1,\ 1]` such that `sin^(-1)x+sin^(-1)y+sin^(-1)z=(3pi)/2` . Find the value of `x^(2016)+y^(2018)+z^(2020)-9/(x^(2016)+y^(2018)+z^(2020))`

Explanation

Given that,

`x , y , z in [-1,1]`

`sin^(-1)x+sin^(-1)y+sin^(-1)z=-(3pi)/2`

`-frac{pi}{2} le sin^(-1) x le frac{pi}{2}`

`sin^(-1) x=sin^(-1) y=sin^(-1) z=-frac{pi}{2}`

`sin(sin^(-1) x)=sin(-frac{pi}{2})`

`x=-1, y=-1, z=-1`


`x^2016+y^2018+z^2020-9/(x^(2016)+y^(2018)+z^(2020))=(-1)^2016+(-1)^2018+(-1)^2020-9/((-1)^(2016)+(-1)^(2018)+(-1)^(2020))=1+1+1-frac{9}{1+1+1}=3-frac{9}{3}=3-3=0`


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