If x=r A B, y=r A B and z=r A , prove that x^2+y^2+z^2=r^2 - Vidyadip

Question

$
\text { If } \mathrm{x}=\mathrm{r} \sin \mathrm{A} \cos \mathrm{B}, \mathrm{y}=\mathrm{r} \sin \mathrm{A} \sin \mathrm{B} \text { and } \mathrm{z}=\mathrm{r} \cos \mathrm{A} \text {, prove that } x^2+y^2+z^2=r^2
$

✓

Answer

$
\begin{aligned}
& \text { LHS }=(r \sin A \cos B)^2+(r \sin A \sin B)^2+(r \cos A)^2 \\
& \Rightarrow r^2 \sin ^2 A \cos ^2 B+r^2 \sin ^2 A \sin ^2 B+r^2 \cos ^2 A \\
& \Rightarrow r^2 \sin ^2 A\left(\cos ^2 B+\sin ^2 B\right)+r^2 \cos ^2 A \\
& \Rightarrow r^2\left(\sin ^2 A+\cos ^2 A\right)=r^2=\text { RHS }
\end{aligned}
$

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