L{sin22t} <------ This was the question

sin2A = 1/2(1-cos2A) <------ this was a note on the question

Below is me trying to laplace it

1/2(1-cos2(2t))

1/2(1-cos4t)

1/2 * 1 - 1/2 cos4t

L 1/2 {1} - L 1/2 { s }
..........s ............s2+32

1 - ....... s .
2s .....2s2+32

as you can see something doesn't seem correct and the answer doesn't flow like the other questions did so im not sure where im going wrong and help would be appreciated.
post looks like a bit of a mess because im not sure how to put in spaces. please ignore the(.)

\displaystyle\begin{aligned}\mathcal{L}\left\{\sin ^2 2t\right\}&=\mathcal{L}\left\{\frac{1}{2}\cdot 1-\frac{1}{2}\cos 4t\right\}\\&=\frac{1}{2}\mathcal{L}\left\{1\right \}-\frac{1}{2}\mathcal{L}\left\{\cos 4t\right\}\\&=\frac{1}{2}\;\frac{1}{s}-\frac{1}{2}\frac{s}{s^2+16}\\&=\frac{1}{2s}-\dfrac{s}{2s^2+32}\end{aligned}

The way I do it is to do "Go Advanced" and then there's a button labeled $\Sigma$ which puts in these bracket-TEX things that go around the equations. Entering the equations is easy enough - there's lots of help online. If you mouse over an equation on MHF it gives you the code used to generate it. You can also look online - google LaTeX and the thing you want to do.