<s>[INST] how is that expression is generated 
I have no idea that how is red arrowed mammoth term is generated from the yellow arrowed term
Please explain
 [/INST] This is just a substitution. You have from Taylor Series
$$
\sin(t) = t- \dfrac{t^3}{3!} + \dfrac{t^5}{5!}- \dfrac{t^7}{7!} + \cdots= t- \dfrac{t^3}{6} + \dfrac{t^5}{120}- \dfrac{t^7}{5040} + \cdots
$$
But you know also the Taylor Series for $e^x-1$:
$$
e^x-1 = x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots
$$
which the book says to call this $t$. But then we plop this Taylor Series where we see $t$ in the Taylor Series for $\sin t$:
$$
\begin{split}
\sin(e^x-1)&= \sin(t) \\
&= t- \dfrac{t^3}{3!} + \dfrac{t^5}{5!}- \dfrac{t^7}{7!} + \cdots \\
&= t- \dfrac{t^3}{6} + \dfrac{t^5}{120}- \dfrac{t^7}{5040} + \cdots \\
&=\left(x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\right)- \dfrac{\left(x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\right)^3}{6} + \dfrac{\left(x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\right)^5}{120}- \dfrac{\left(x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\right)^7}{5040} + \cdots
\end{split}
$$
</s>