यदि `cos^(-1)x + cos^(-1)y + cos^(-1) z = x`,
सिद्ध करे कि `x^(2) + y^(2) + z^(2) + 2xyz = 1.`

Explanation

माना कि, `cos^(-1) x = alpha, cos^(-1) y = beta` तथा `cos^(-1) z = gamma `
`rArr cos alpha = x, cos beta = y` तथा `cos gamma = z `
प्रश्न से, `alpha + beta + gamma = pi therefore alpha + beta = pi -gamma`
`therefore cos (alpha + beta) = cos (pi-gamma)`
या, `cos alpha cos beta - sin alpha sin beta = - cos gamma`
या, `xy-sqrt(1-x^(2)) sqrt(1-y^(2)) = -z` या, `xy + z = sqrt(1-x^(2)) sqrt(1-y^(2))`
दोनों तरफ करने पर, `x^(2) y^(2) + z^(2) + 2xyz = 1 - x^(2) - y^(2) + x^(2) y^(2)`
या, `x^(2) + y^(2) + z^(2) + 2xyz = 1`