When faced with an exponent containing a $\pm$, how is it handled/simplified?
For example, this expression: $$e^{\pm i\omega\sqrt{1-\zeta^2}t}\qquad(1)$$
Expands to: $$ c_1\,e^{i\omega\sqrt{1-\zeta^2}t}+c_2\,e^{-i\omega\sqrt{1-\zeta^2}t} \qquad (2)$$
And this becomes: $$ A\,cos(\sqrt{1-\zeta^2}\omega t)+B\,sin(\sqrt{1-\zeta^2}\omega t)\qquad (3)$$
However, it is unclear to me how the $\pm$ sign led to this breakdown from $(1)$ to $(2)$ and how it was further broken down with the Euler identity and arbitrary constants into $(3)$.
What steps happened in-between expressions to progressively massage these terms, and how are they explained/justified?
P.S.: If anyone is wondering, these terms came from part of a solution of a second-order homogeneous linear differential equation in a vibrations/structural dynamics textbook.