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)?#

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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)?#

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Answer 1 · 2016-08-23T13:00:56

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)?#

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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)?#

1 Answer

Explanation:

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