Answer
$x^2+y^2+z^2=R^2$
Work Step by Step
Start with the left side:
$x^2+y^2+z^2$
Substitute $x=R\cos\theta\sin \phi$, $y=R\sin\theta\sin\phi$, and $z=R\cos\phi$:
$=(R\cos\theta\sin\phi)^2+(R\sin\theta\sin\phi)^2+(R\cos\phi)^2$
$=R^2\cos^2\theta\sin^2\phi+R^2\sin^2\theta\sin^2\phi+R^2\cos^2\phi$
Factor out $R^2$:
$=R^2(\cos^2\theta\sin^2\phi+\sin^2\theta\sin^2\phi+\cos^2\phi)$
Factor out $\sin^2\phi$ from the first two terms:
$=R^2(\sin^2\phi(\cos^2\theta+\sin^2\theta)+\cos^2\phi)$
$=R^2(\sin^2\phi*1+\cos^2\phi)$
$=R^2(\sin^2\phi+\cos^2\phi)$
$=R^2*1$
$=R^2$
Since this equals the right side, the identity has been proven.
