Infinitesimal displacement in spherical coordinates

[PWiz]

I'm trying to derive what ##ds^2## equals to in spherical coordinates.
In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$
Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle)
For simplicity, let ##cos \phi = A## and ##sin \phi = B##
Then ##dx=r \ cos\theta \ dB + B(cos \theta \ dr - r \ sin \theta \ d \theta)##
##dy=r \ sin\theta \ dB + B(sin \theta \ dr + r \ cos \theta \ d \theta)##
##dz= r \ dA + A \ dr##
When I add up the squares of the 3 differential elements above, gather similar terms and replace ##A## and ##B##, I get ##ds^2 = dr^2 + r^2 \ sin^2 \phi \ d \theta^2## , and you can see that one term is missing. Is something wrong with my approach?

 

[PWiz]

$$ z = r\cos{\theta} $$

I'm using ##\phi ## as the polar angle and ##\theta## as the azimuthal angle over here. Either way, that won't add the missing term, but will only interchange the angle variables.

 

[theodoros.mihos]

Sorry I see later.
$$ x = rB\cos{theta} \,\Rightarrow\, dx = dr\,B\cos{\theta} + r\,dB\,\cos{theta} - rB\sin{\theta} = (B\,dr + r\,dB)\cos{\theta} - rB\sin{\theta} $$
$$ dy = (B\,dr + r\,dB)\sin{\theta} +rB\cos{\theta} \,\text{and}\, dz = A\,dr-rB\,d\phi $$
because
$$ A = cos{\phi} \Rightarrow dA = -\sin{\phi}\,d\phi = -B\,d\phi \,\text{and}\, dB = A\,d\phi $$
I think your problem is that A depends by B.

 

[PWiz]

I found an error in my steps. I'm getting the correct answer now. Anyways, thanks for helping :)