Nth derivative of a trignometric function

[bluevires]

The question is uploaded as an attachment.

By looking at the question, I can see that the number of terms of the derivative of this function is increasing exponentially, but since there's a product involved, I'm having problem finding a pattern..But i can see it has something to do with the odd/eveness of the order of derivative.

Any help would be appreciated

 

[z-component]

What usually helps with these problems is writing out the first few derivatives of the function. You should be able to notice the pattern, but until you do, keep differentiating.

 

[HallsofIvy]

Notice that you are only asked for the value of that derivative at x= 0. That pattern might be much easier to spot (and prove) than the general derivative.