Hi, I'm aditya. Nice to meet you. I have a problem with vibration derivation.

I don't know the answer,

How could possibly the trigonometric function of

become the form

Any chance that the former can be the latter form? I've searched and looked thorough and still don't have the answer until now.

The list of what I've been through is

List of trigonometric identities - Wikipedia, the free encyclopedia
Abramowitz and Stegun, p. 80, 4.4.42

Stumbled upon my own hand-held (writing) derivation which was not satisfactory.

I just don't know how to solve this problem. I think it's a mathematics one. So i tried to submit the problem via this math Forum in there is possibility of someone solving "the unsolved".

It was originally a physics problem. The Vibration derivation of This problem is as follows (it is from the book of Singiresu Rao, Mechanical Vibrations).

Thanks. Any Help would be very appreciated