Question

If cos^-1 x + cos ^-1 y + cos ^-1 z = π , prove that x² + y² + z² + 2xyz = 1.

Answer 1

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