If $t=\tanh\frac{x}{2}$, prove that $\sinh x = \frac{2t}{1-t^2}$ and $\cosh x = \frac{1+t^2}{1-t^2}$. 
If $t=\tanh\frac{x}{2}$, prove that $\sinh x = \frac{2t}{1-t^2}$ and $\cosh x = \frac{1+t^2}{1-t^2}$. Hence solve the equation $7\sinh x + 20 \cosh x = 24$.

I have tried starting by writing out $\tanh\frac{x}{2}$ in exponential form and then squaring it but I can't make any progress from this.

Note that
(i) tanh($x$) = sinh(x) / cosh($x$)
(ii) 1 - tanh$^2(x$) = 1/ cosh$^2(x$)
(iii) sinh($x + y$) = sinh($x$)cosh($y$) + sinh($y$)cosh($x$)
and apply these to the term
$\frac{2t}{1 - t^2}$ with $t =$ tan($x/2$) to directly obtain your result for sinh($x$). The formula for cosh$(x)$ follows then directly by applying (i) again.
