Question

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

Answer 1

$x=rcos\theta cos\varphi ;y=rcos\theta sin\varphi ;z=rsin\theta$

$x^2+y^2+z^2=r^2.$

$x^2=r^{2}co{s}^2\theta co{s}^2\varphi$

$y^2=r^{2}co{s}^2\theta si{n}^2\varphi$

$z^2=r^{2}si{n}^2\theta .$

$r^{2}co{s}^2\theta co{s}^2\varphi +r^{2}co{s}^2\theta si{n}^2\varphi +r^{2}si{n}^2\theta =r^2$

$r^2(co{s}^2\theta co{s}^2\varphi +co{s}^2\theta si{n}^2\varphi +si{n}^2\theta )=r^2$

$r^2\left[co{s}^2\theta \right(si{n}^2\varphi +co{s}^2\varphi )+si{n}^2\theta ]=r^2$

$r^2\left[co{s}^2\theta \right(1)+si{n}^2\theta ]=r^2$

$r^2\left(1\right)=r^2$

$\therefore r^2=r^2$

$\therefore$ Hence Proved $x^2+y^2+z^2=r^2$