Answer :

Given: x = r sin θ cos ϕ, y = r sin θ sin φ and z = r cos θ, 

Solution:

x = r sin θ cos ϕ

Squaring both sides, we get

x2 = r2 sin2 θ cos2ϕ ……….(i)

and y = r sin θ sin ϕ

Squaring both sides, we get

y2 = r2 sin2 θ sin2ϕ ……….(ii)

z = r cos θ

Squaring both sides, we get

z2 = r2 cos2 θ ……….(iii)

Adding (i), (ii) and (iii), we get

x2 + y2 + z2 = r2 sin2 θ cos2ϕ + r2 sin2 θ sin2ϕ + r2 cos2 θ

= r2 (sin2 θ cos2ϕ + sin2 θ sin2ϕ + cos2 θ)

= r2 [sin2 θ (cos2ϕ + sin2ϕ) + cos2 θ]

∵ sin2 θ + cos2 θ = 1

= r2 [sin2 θ + cos2 θ] 

 Again apply the identity  sin2 θ + cos2 θ = 1

= r2 

Hence x2 + y2 + z2 = r2

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