Well, i am reading the book Elliptic Theory on singular manifolds, Vladimir E. Nazaikinskii Anton Yu. Savin Bert-Wolfgang Schulze Boris Yu. Sternin,
The manifold is the cone given by $x^{2}+y^{2}=z^{2}\sin^{2}\theta$, with $z\geq 0$, where $\theta$ is the angle between axis and the generator of the cone.
I am not expert in this subject, i am learning, so i try with the simple case, if i suppose that
we are workimg with $x^{2}+y^{2}=z^{2}$ then i put $f(x,y)=(x,y,\sqrt{x^{2}+y^{2}})$. Hence, if i choose $x=r\cos\theta$ and $y=r\sin\theta$ hence
$dx=dr\cos\theta -r\sin\theta d\theta$
$dy= dr\sin\theta +r\cos\theta d\theta$
$dz=dr$
Then, when i compute $dx^{2}+dy^{2}+dz^{2}= 2dr^{2}+r^{2}d\theta^{2}$, but now, if i tried the same is turn complicate because $z=(\sin\theta)^{-1}\sqrt{x^{2}+y^{2}}$ and $\varphi$ is the polar angle on the plane $xy$ and $r$ is the distance betweeen vertex and the point that i will choose in the cone. I did not understand the line $r^{2}=x^{2}+y^{2}+z^{2}=(x^{2}+y^{2})(1+\sin^{2}\theta)$ above in the picture, is a mistake?
Please somebody can to help me or give me a hint please, i want to proof that
$d_s^{2}= d_r^{2}+r^{2}\sin^{2}\theta d_\varphi^{2}$.
After of the hint from Svyatoslav, i have my firs relation
$\sin\theta = \frac{\sqrt{x^2+y^2}}{r}$, and $\cos\theta= \frac{z}{r}$. On the plane $xy$ i have $\cos \varphi=\frac{x}{\sqrt{x^2+y^2}}$ and $\sin\varphi=\frac{y}{\sqrt{x^2+y^2}}$, hence
$x=\sqrt{x^2+y^2}cos\varphi=r\sin\theta\cos\varphi$ and
$y=\sqrt{x^2+y^2}sin\varphi=r\sin\theta\sin\varphi$, now when i fix $\theta$ implies
$dx= \sin\theta (dr\cos\varphi -r\sin\varphi d\varphi)$,
$dy= \sin\theta(dr\sin\varphi + r\cos\varphi d\varphi)$. So,
$dx^2= \sin^2\theta(dr^2\cos^2\varphi -2 rdrd\varphi\sin\varphi\cos\varphi + r^2\sin^2\varphi d\varphi^2)$,
$dy^2= \sin^2\theta(dr^2\sin^2\varphi + 2 rdrd\varphi\sin\varphi\cos\varphi + r^2\cos^2\varphi d\varphi^2 )$,
$dz^2=dr^2\cos^2\theta$, finally
$dx^2+dy^2+dz^2= dr^2 +r^2\sin^2\theta d\varphi^2$.