In order to multiply two power series, say
\begin{align*}
\def\bl#1{\color{blue}{#1}}
\def\gr#1{\color{green}{#1}}
\bl{A}(x) &= \bl{a_0} + \bl{a_1}x + \bl{a_2}x^2 + \cdots \\
\gr{B}(x) &= \gr{b_0} + \gr{b_1}x + \gr{b_2}x^2 + \cdots,
\end{align*}
we have to imagine opening parentheses:
\begin{align*}
\bl{A}(x) \, \gr{B}(x)
&= \bigl( \bl{a_0} + \bl{a_1}x + \bl{a_2}x^2 + \cdots \bigr) \,
\bigl( \gr{b_0} + \gr{b_1}x + \gr{b_2}x^2 + \cdots \bigr) \\
&= \bl{a_0} \,
\bigl( \gr{b_0} + \gr{b_1}x + \gr{b_2}x^2 + \cdots \bigr) \\
&\quad {}+ \bl{a_1}x \,
\bigl( \gr{b_0} + \gr{b_1}x + \gr{b_2}x^2 + \cdots \bigr) \\
&\qquad {}+ \bl{a_2}x^2 \,
\bigl( \gr{b_0} + \gr{b_1}x + \gr{b_2}x^2 + \cdots \bigr) \\
&\qquad\quad {}+\cdots \\
&= \bigl( \bl{a_0}\gr{b_0} + \bl{a_0}\gr{b_1}x
+ \bl{a_0}\gr{b_2}x^2 + \cdots \bigr) \\
&\quad {}+ \bigl( \bl{a_1}\gr{b_0}x
+ \bl{a_1}\gr{b_1}x^2 + \bl{a_1}\gr{b_2}x^3 + \cdots \bigr) \\
&\qquad {}+ \bigl( \bl{a_2}\gr{b_0}x^2 + \bl{a_2}\gr{b_1}x^3
+ \bl{a_2}\gr{b_2}x^4 + \cdots \bigr) \\
&\qquad\quad {}+\cdots
\end{align*}
Now, we collect like terms:
\begin{align*}
\bl{A}(x) \, \gr{B}(x)
&= \bigl( \bl{a_0}\gr{b_0} \bigr) \\
&\quad {}+ \bigl( \bl{a_0}\gr{b_1} + \bl{a_1}\gr{b_0} \bigr) x \\
&\qquad {}+ \bigl( \bl{a_0}\gr{b_2} + \bl{a_1}\gr{b_1}
+ \bl{a_2}\gr{b_0} \bigr) x^2 \\
&\qquad\quad {}+\cdots
\end{align*}
In general, the $x^n$ coefficient in $\bl{A}(x) \gr{B}(x)$ is
$$
\bl{a_0}\gr{b_n} + \bl{a_1}\gr{b_{n-1}}
+ \cdots + \bl{a_{n-1}}\gr{b_1} + \bl{a_n}\gr{b_0},
$$
i.e. a sum of terms $\bl{a_i}\gr{b_j}$,
where $\bl{i} + \gr{j} = n$.
Can you see how to use these observations to square the series for the hyperbolic trig. functions? There are some rather straightforward patterns that you should recognize before having to compute too many coefficients. Try it!