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Question
If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.
Solution
0 ≤ cos-1x ≤ π and
cos-1x + cos-1 y + cos-1z = 3π
∴ cos-1x = π, cos-1y = π and cos-1z = π
∴ x = y = z = cos π = - 1
∴ x2 + y2 + z2 + 2xyz
= (- 1)2 + (- 1)2 + (- 1)2 + 2(- 1)(- 1)(- 1)
= 1 + 1 + 1 - 2
= 3 - 2
= 1.
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