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Question
If x = r sin θ cos Φ, y = r sin θ sin Φ and z = r cos θ, prove that x2 + y2 + z2 = r2.
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Solution
We have,
x = r sin θ cos Φ,
y = r sin θ sin Φ,
z = r cos θ
Squaring and adding,
x2 + y2 + z2
= r2 sin2θ cos2Φ + r2 sin2θ sin2Φ + r2 cos2θ
= r2 sin2θ (cos2Φ + sin2Φ) + r2 cos2θ
= r2 sin2θ x (1) + r2 cos2θ
= r2 (sin2θ + cos2θ)
= r2 x 1 = r2
Hence, x2 + y2 + z2 = r2.
Hence proved.
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