I have the following expression $$\frac 12 x_0e^{-\beta t}\left[\left(\frac {\beta}{i \sqrt{\omega ^2-\beta ^2}}+1\right)e^{i \sqrt{\omega ^2 - \beta ^2}t}+\left(\frac {- \beta}{i \sqrt{\omega ^2 - \beta ^2}}+1\right)e^{-i \sqrt{\omega ^2 - \beta ^2}t}\right]$$
and need to write it in the following form $$x(t)=Ae^{- \beta t}cos(\omega t+ \phi)$$
I know I need to use the rule that $cos(\theta )=\frac 12 (e^{i \theta} + e^{-i \theta})$ but the extra minus is confusing me. I've checked back through all my previous work and am pretty certain it's supposed to be there, but it messes with my simplification. Help!
(Edited to fix the wrong fraction at the beginning - shouldn't really make a difference)
The complex unit i in the denominator should give you more cause for consideration.
Your expression is of the form, leaving out the leading exponential,
$$α e^{iλ t}+\bar α e^{-iλ t}=2\, Re(α e^{iλ t})=2|α|\,Re(e^{iλ t+i\arg(α)}).$$
The announced form of the result is not obtainable if $ω$ means the same number in all occurrences.