Answer
$\dfrac{2 \sqrt 2(s+1)}{(s^2+1)}$
Work Step by Step
The Laplace Transform can be written as:
$L[F(t)]=\int_{0}^{\infty} e^{-st} f(t) dt $
We are given that $f(t)=4 \cos (t-\dfrac{\pi}{4})$
Now, $L[F(t)]=4 L[\cos (t-\dfrac{\pi}{4})] \\=4 L[\cos t \cos (\pi/4)-\sin t+\sin (\pi/4)] \\=\dfrac{(4)(s)}{\sqrt 2(s^2+1)}+\dfrac{(4)(s)}{\sqrt 2(s^2+1)}\\=\dfrac{2 \sqrt 2(s+1)}{(s^2+1)}$
