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. 

Answer 1

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.