Question

If $x=r\cos \theta .\sin \varphi ,y=\sin \theta .\sin \varphi ,z=r\cos \varphi$ Prove that $x^2+y^2+z^2=r^2$

Answer 1

Given, $x=r\cos \theta sin\varphi$

$y=r\sin \theta sin\varphi$

$z=rcos\varphi$

to prove that $x^2+y^2+z^2=r^2$

LHS $=x^2+y^2+z^2$

putting value of $x,y$ and $z$

$\Rightarrow r^2{\cos }^2\theta {\sin }^2\varphi +r^2{\sin }^2\theta {\sin }^2\varphi +r^2{\cos }^2\varphi$

$\Rightarrow r^2{\sin }^2\varphi ({\cos }^2\theta +{\sin }^2\theta )+r^2{\cos }^2\varphi$

$\Rightarrow r^2{\sin }^2\varphi +r^2{\cos }^2\varphi$

$\Rightarrow r^2({\sin }^2\varphi +{\cos }^2\varphi )\because {\sin }^{2}x+{\cos }^{2}x=1$

$\Rightarrow r^2=RHS$ Hence proved