Q47 of 268 Page 5

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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