How do I prove the Cos formula?

Answer 1

I am assuming you mean the derivative of the cosine function

The difference quotient argument will not work because the limit involves the

using L'Hopital's rule, which in turn requires the derivative of the cosine, which

we are trying to prove.

Instead, the MacLaurin series expansion for cosine is:

1 - x^2/2 + x^4/4! - x^6/6! + x^8/8! + .....

The derivative of this MacLaurin polynomial is:

-x + x^3/3! - x^5/5! + x^7/7! + ----

factoring out the negative sign gives:

-(x - x^3/3! + x^5/5! - x^7/7! + ----

which turns out to be the opposite of the MacLaurin series expansion of the sine function)

Examining the general terms of these series, cosine is:

(-1)^n/(2n)! x^(2n), n=0,1,2,3,....

the derivative is:

(-1)^n/ (2n-1)!) x^(2n-1)

which is the general term for the MacLaurin series of the sine function, but with

the loop counter decreased