If x= r cosα sinβ, y = r sinα sinβ and z = r cosα then prove that

1.	If x= r cosα sinβ, y = r sinα sinβ and z = r cosα then prove that x2 + y2 + z2 = r2.
Answer» Taking LHS = x² + y² + z² Putting the values of x, y and z , we get =(r cos α sin β)² + (r sin α sin β)² + (r cos α)² = r² cos²α sin²β + r² sin²α sin²β + r² cos²α Taking common r² sin² α , we get = r² sin²α (cos²β + sin² β) + r²cos² α = r² sin²α + r² cos² α [∵ cos² β + sin² β = 1] =r² ( sin² α + cos² α) = r² [∵ cos² α + sin² α = 1] = RHS Hence Proved

If x= r cosα sinβ, y = r sinα sinβ and z = r cosα then prove that x2 + y2 + z2 = r2.

Answer»

Taking LHS = x² + y² + z²

Putting the values of x, y and z , we get

=(r cos α sin β)² + (r sin α sin β)² + (r cos α)²

= r² cos²α sin²β + r² sin²α sin²β + r² cos²α

Taking common r² sin² α , we get

= r² sin²α (cos²β + sin² β) + r²cos² α

= r² sin²α + r² cos² α [∵ cos² β + sin² β = 1]

=r² ( sin² α + cos² α)

= r² [∵ cos² α + sin² α = 1]

= RHS

Hence Proved