Question

How do you prove that `(x+pi/4)-(x+pi/4)^3/(3!)+(x+pi/4)^5/(5!)-... = 1-(x-pi/4)^2/(2!)+(x-pi/4)^4/(4!)=...= 1+x-x^2/(2!)-x^3/(3!)+x^4/(4!)+x^5/(5!)-..., x in (-oo, oo)?`

Answer 1

See below

Explanation:

`sin(x+pi/4) = cos(x-pi/4)` and

`sin(x+pi/4) = sqrt(2)/2(sin x + cos x)`

so the ask would be

`(x+pi/4)-(x+pi/4)^3/(3!)+(x+pi/4)^5/(5!)-cdots = 1-(x-pi/4)^2/(2!)+(x-pi/4)^4/(4!)= sqrt(2)/2(1+x-x^2/(2!)-x^3/(3!)+x^4/(4!)+x^5/(5!)-cdots), x in (-oo, oo)?`