# Transform: displaystyle f{{left({t}right)}}={{cos}^{2}{a}}{t}

Transform:
$f\left(t\right)={\mathrm{cos}}^{2}at$
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Step 1
The function is $f\left(t\right)={\mathrm{cos}}^{2}at$
Use the identity ${\mathrm{cos}}^{2}x=\frac{1+{\mathrm{cos}}^{2}x}{2}$
Rewrite the function in a linear form.
$f\left(t\right)=\frac{1+\mathrm{cos}\left(2at\right)}{2}$
$=\frac{1}{2}+\frac{1}{2}\mathrm{cos}\left(2at\right)$
Step 2
Use to following Laplace transformations.
$L\left\{1\right\}=\frac{1}{s}$
$L\left\{\mathrm{cos}\left(kt\right)\right\}=\frac{s}{{s}^{2}+{k}^{2}}$
Determine the Laplace transformation.
$L\left\{{\mathrm{cos}}^{2}\left(at\right)\right\}=\frac{1}{2}L\left\{1\right\}+12L\left\{\mathrm{cos}\left(2at\right)\right\}$
$=\frac{1}{2}\left(\frac{1}{s}\right)+\frac{1}{2}\frac{s}{{s}^{2}+{\left(2a\right)}^{2}}$
$=\frac{1}{2}\left(\frac{1}{s}+\frac{s}{{s}^{2}+4{a}^{2}}\right)$
$=\frac{1}{2}\left(\frac{{s}^{2}+4{a}^{2}+{s}^{2}}{s\left({s}^{2}+4{a}^{2}\right)}\right)$
$=\frac{1}{2}\left(\frac{2{s}^{2}+4{a}^{2}}{s\left({s}^{2}+4{a}^{2}\right)}\right)$
$=\frac{{s}^{2}+2{a}^{2}}{s\left({s}^{2}+4{a}^{2}\right)}$
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