Laplace Transform of Cosine and Hyperbolic Cosine

[chickens]

Homework Statement

L[cos(at)cosh(at)] = ?

Homework Equations

L[cos(at)] = s/(s^2 + w^2)

The Attempt at a Solution

I'm able to get the solution given that is (s^3)/(s^4 + 4a^4)

The question requested to use first shift property. So I used cosh(at) = 1/2[e^at + e^(-at)]

but now I'm trying to use another method.

if you use euler's rule, you can also get cos(at) = cosh(at) = 1/2[e^at + e^(-at)]

then simply multiply them and get L[e^2at + 2 + e^(-2at)] but I don't seem to get the same solution as above, any ideas?

 

[HallsofIvy]

Homework Statement

L[cos(at)cosh(at)] = ?

Homework Equations

L[cos(at)] = s/(s^2 + w^2)

The Attempt at a Solution

I'm able to get the solution given that is (s^3)/(s^4 + 4a^4)

The question requested to use first shift property. So I used cosh(at) = 1/2[e^at + e^(-at)]

but now I'm trying to use another method.

if you use euler's rule, you can also get cos(at) = cosh(at) = 1/2[e^at + e^(-at)]

No, Euler's rule does NOT say cos(at)= cosh(at)!
You may be thinking of
$$cos(at)= \frac{e^{iat}+ e^{-iat}}{2}$$
Note the "i" s in that!

then simply multiply them and get L[e^2at + 2 + e^(-2at)] but I don't seem to get the same solution as above, any ideas?