We start with the definition of the Laplace Transform L{f(t)} which is represented by the function F(s).
F(s)=∫e⁻ˢᵗf(t)dt where the integral is to be evaluated for t between 0 and ∞. Let f(t)=cos(bt).
Let u=e⁻ˢᵗ so du=-se⁻ˢᵗ and let dv=cos(bt)dt so v=sin(bt)/b. Integrating by parts:
F(s)=e⁻ˢᵗsin(bt)/b+(s/b)∫e⁻ˢᵗsin(bt)dt.
If we let dv=sin(bt)dt, then v=-cos(bt)/b, and we can integrate the above integral by parts:
∫e⁻ˢᵗsin(bt)dt=-e⁻ˢᵗcos(bt)/b-(s/b)∫e⁻ˢᵗcos(bt)dt, but the latter integral is identical to F(s) so:
F(s)=e⁻ˢᵗsin(bt)/b-(s/b²)e⁻ˢᵗcos(bt)-(s/b)²F(s).
Therefore, F(s)(1+(s/b)²)=e⁻ˢᵗsin(bt)/b-(s/b²)e⁻ˢᵗcos(bt).
We now apply the limits 0 and ∞: F(s)(1+(s/b)²)=0-(-s/b²)=s/b².
Multiply through by b²: F(s)(s²+b²)=s, so F(s)=L{cos(bt)}=s/(s²+b²) QED.