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Question
If cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.
Solution
cos-1 x + cos -1 y + cos -1 z = π
cos-1 x + cos -1 y = π - cos -1 z
cos-1 `(xy - sqrt(1 - x^2) sqrt(1 -y^2 ))` = π - cos -1 z
`xy - sqrt(1 - x^2) sqrt(1 -y^2)` = cos ( π - cos -1 z)
`xy - sqrt(1 - x^2) sqrt(1 -y^2)` = - cos(cos-1 z)
xy - `sqrt(1 - x^2) sqrt(1 -y^2) = -z`
`xy + z = sqrt(1 - x^2) sqrt(1 - y^2)`
Squaring both sides, we have
(xy + z)2 = (1 - x2) (1- y2)
x2y2 + z2 + 2xyz = 1 - x2 - y2 + x2y2
x2 + y2 + z2 + 2xyz = 1
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