If ` x=r sin alpha cos beta, y= r sin alpha sin beta and z= r cos alpha , ` prove that
` x^(2)+y^(2) + z^(2) = r^(2). `
` x^(2)+y^(2) + z^(2) = r^(2). `
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We have
` x^(2) + y^(2) + z^(2) = r^(2) sin^(2) alpha cos^(2) beta +r^(2) sin^(2) alpha sin^(2) beta + r^(2) cos^(2) alpha `
` = r^(2)sin^(2) alpha (cos^(2) beta + sin^(2) beta) + r^(2) cos^(2) alpha `
`= r^(2) sin^(2) alpha + r^(2) cos^(2) alpha ” ” [because cos^(2) beta + sin^(2) beta =1] `
`= r^(2) (sin^(2) alpha + cos^(2) alpha )= r^(2) ” ” [because sin^(2) alpha + cos^(2) alpha =1]. `
Hence, ` (x^(2) + y^(2) + z^(2))= r^(2) . `