I am confused by the following (possibly simple) algebra in the proof of the Cantelli-Cheybyshev inequality. I am following Rohatgi and Saleh (2015, Section 3.4 Lemma 1) where we plug in $\phi(x)=(x+c)^2$ into Markov's inequality to get:
\begin{equation}
P(Y \geq t) \leq P\big( \phi(Y) \geq (t+c)^2 \big) \leq \underbrace{\frac{{E}[(Y+c)^2]}{(t+c)^2}}_{RHS}
\end{equation}
where $EY=0, t\geq 1,$ and $c>0$. To find the tightest bound, we minimize the RHS wrt to $c$. This gives us
\begin{equation}
c^* = \frac{Var Y}{t}
\end{equation}
The proof in Rohatgi and Saleh (2015) and elsewhere (e.g. Cantelli's inequality proof) says plugging in $c^*$ gives the desired expression:
\begin{equation}
P(Y \geq t) \leq \frac{Var Y}{Var Y + t^2}
\end{equation}
But the algebra in between doesn't seem trivial (at least to me): if I plug in $c^*$ into RHS, I don't see how to get the RHS of Cantelli's inequality even knowing that I can add or substract $EY=0$:
\begin{equation}
\frac{{E}[(Y+c)^2]}{(t+c)^2}\bigg|_{c=c^*} = \frac{Var Y + (Var Y/t)^2}{t^2 + 2t(Var Y/t) + (Var Y/t)^2} = ...?
\end{equation}
Any help would be greatly appreciated!
