Find the Fourier cosine coefficients of

Differentiating yields:

,

the Fourier sine series of e^x. Differentiating again yields

,

Since equations 1 and 3 both give Fourier cosine series of e^x, they must be identical. Thus,

and

Can anyone please explain step by step what is wrong with this? I'm supposed to correct the mistakes and then find without using the typical technique but I'm so confused!

Any mistakes at all you can see, please point them out!

Thanks in advance