The [tex]i[/tex] in the denominator is missing in the second expression. It should be
[tex]\sin at=\dfrac{e^{iat}-e^{-iat}}{2i}[/tex]
This follows from
[tex]e^{iat}=\cos at+i\sin at[/tex]
[tex]e^{-iat}=\cos at-i\sin at[/tex]
[tex]\implies e^{iat}-e^{-iat}=2i\sin at[/tex]
[tex]\implies \sin at=\dfrac{e^{iat}-e^{-iat}}{2i}[/tex]
