# If x= r cos α sin β, y = r sin α sin β and z = r cos α then prove that x2 + y2 + z2 = r2.

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

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