Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates.
I used the facts that
$$ \begin{align} x&=ρ\sin\theta\cos\phi\;,\\ z&=ρ\cos\phi\;, \end{align} $$
And ended up with:
$ 4 (ρ^2 \sin^2(\phi) \cos(\theta) + ρ^2\cos^2(\phi))= 5 $
But it's not simplified enough? I can't use Pythagoras' theorem: $\cos^2(\theta) + \sin^2(\theta) = 1$ inside the parenthesis.
So what do?
It depends how you define spherical coordinates that ultimately determines what the equation looks like. If you define spherical coordinates like this
$$x = \rho \sin \theta \cos \phi$$ $$y = \rho \sin \theta \sin \phi$$ $$z = \rho \cos \theta$$
where $\rho$ is the radius, $\theta$ is the inclination, and $\phi$ is the azimuth. Naturally, $\rho \in [0, \infty), \> \theta \in [0, \pi], \> \phi \in [0, 2 \pi]$.
(notice that the cosines are different for x and z unlike in your definition, although I think this is what you meant)
then
$$\rho^2(\cos^2 \phi \sin^2 \theta + \cos^2 \theta) = \frac{5}{4} $$ $$\rho^2(\cos^2 \phi \sin^2 \theta + 1 - \sin^2 \theta) = \frac{5}{4} $$ $$\rho^2(\sin^2\theta(\cos^2 \phi -1) + 1) = \frac{5}{4}$$ $$\rho^2(1 - \sin^2 \theta \sin^2 \phi) = \frac{5}{4}$$
is the simplest. However, you do not have to define spherical coordinates in such a manner.
This convention is used more often in physics where in mathematics you would often see $\theta$ and $\phi$ be flipped in definition.