This is homework,so no answers please.
Prove that the topology on $RP^{n}$ given by the standard smooth structure (lines through the origin in $\mathbb{R}^{n+1}/\{0\}$ and $\tau_{1}$) is equal to the quotient topology induced by ∼ from the standard topology on $S^{n}$ ($\tau_{2}$) (identifying antipodal points).
The standard smooth structure is: The chart from $\mathbb{R}P^{n}$ to $\mathbb{R}^{n}$ is $\phi_{i}[x_{1},...,x_{n}]=(\frac{x_{1}}{x_{i}},...,\hat{1},...,\frac{x_{n}}{x_{i}})$ and $U_{i}=\{x\in \mathbb{R}^{n+1}/\{0\}:x_{i}\neq 0\}$. The induced topology $\tau_{1}$ is: U is open if $\phi_{i}(U)$ is open in $\mathbb{R}^{n}$. The $\phi_{i}(U)$ is the intersection of open cone [U] with affine hyperplane with $x^{i}=1$
I am Not comparing $\tau_{2}$ to the quotient topology of $\mathbb{R}^{n+1}/\{0\}/\sim$, where $x\sim y$ iff $x=\lambda y$.
any mistakes in showing $\tau_{1}=\tau_{2}$:
Given open U in $\mathbb{S}^{n}/\sim$, we have $\pi_{2}^{-1}(U)=U\cup (-U)$ is open in $\mathbb{S}^{n}$. Then via chart $\phi_{a}$ of $\mathbb{S}^{n}$, we get open $V=\phi_{a}(U)$ in $\mathbb{R}^{n}$. Then by appropriate shrinking we get $\tilde{V}=\phi_{i}(U)$ open and so $U \in \tau_{1}$.
Conversely, given $U \in \tau_{1}$, we can reverse the above process to get $U\cup (-U)$ being open (since -U is open). Thus, $U\in \tau_{2}$.
You are working rather hard. Here is a hint: if $f : X \to Q$ is a quotient map and $Q$ is Hausdorff, and if $Y \subset X$ is a compact subset such that $f \mid Y : Y \to Q$ is surjective, then $f \mid Y$ is also a quotient map. Therefore $Q$ can be viewed as a quotient of both $X$ and $Y$. By carefully choosing $X$ and $Y$ and $f$, you should be able to answer your question quite easily.
ADDED: Here's a few more details of the hint. The "topology on $RP^n$ given by the standard smooth structure" is indeed a quotient topology. It is not a quotient of $\mathbb{R}^{n+1}$, but instead it is a quotient of $\mathbb{R}^{n+1}-\{0\}$. Those charts in that atlas you describe are quotient maps from saturated (open) subsets of $\mathbb{R}^{n+1}-\{0\}$ to their (open) images in $RP^n$, and they are also restrictions of the globally defined quotient map $\mathbb{R}^{n+1}-\{0\} \to RP^n$.