Question

Q- IF Xcosø - y sinø = Z
Prove that- Xsinø- ycosø = √x^2+y^2-z^2​

Answer 1

We have

x = r sin θ cos ɸ

y = r sin θ sin ɸ

z = r cos θ

squaring and adding,

x2 + y2 + z2

= r2 sin2 θ cos2 ɸ + r2 sin2 θ sin2 ɸ + r2 cos2 θ

= r2 sin2 θ (cos2 ɸ + sin2 ɸ) + r2 cos2 θ

= r2 sin2 θ × 1 + r2 cos2 θ

= r2 (sin2 θ + cos2 θ)

= r2 × 1 = r2

Hence x2 + y2 + z2 = r2

Hence proved

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