I have seen several different expressions of complex projective spaces $$ \mathbf{CP}^n =\frac{SU(n+1)}{S(U(1) \times U(n ))} \tag{1} $$ $$ \mathbf{CP}^n = \frac{SU(n+1)}{U(1) \times SU(n)} \tag{2} $$ $$ \mathbf{CP}^n = \frac{U(n+1)}{U(1) \times U(n )}\tag{3} $$ Here we can use $U(n)=\frac{SU(n) \times U(1)}{\mathbb{Z}_{n}}$. The (3) becomes $\frac{\frac{SU(n+1) \times U(1)}{\mathbb{Z}_{n+1}}}{U(1) \times \frac{SU(n) \times U(1)}{\mathbb{Z}_{n}}}$. The $U(1)$ can be different factors of $U(n)$, so one needs to define them carefully.
Question: How do we show that all of these (1),(2),(3) are the $\mathbf{CP}^n$? Please firstly define the expressions above. And how do we prove them and how to understand them to be the same (or not) intuitively?
Note added: In comparison, see that Wikipedia uses two expressions mentioned above:
Defining $\mathbb CP^n$ as the space of complex lines in $\mathbb C^{n+1}$, the groups $U(n+1)$ and $SU(n+1)$ evidently act transitively on $\mathbb CP^n$. (Since multiples of the identity act trivially on $\mathbb CP^n$, there is no real gain in using $U$ rather than $SU$, but never mind.) The stabilizer of the line spanned by the first basis vector in these two groups is $U(1)\times U(n)$ (block-diagonal matrices) respectively $S(U(1)\times U(n))$ which leads to the first and last expression that you list. The middle expression in my opinion is not really clear, one would have to specify an embedding of $U(1)\times SU(n)$ into $SU(n+1)$ in order to give it a precise meaning.