Write cos2(x) as linear combination of x↦sin⁡(x) and x↦cos⁡(x)

poveli1e

poveli1e

Answered

2022-01-24

Write cos2(x) as linear combination of xsin(x) and xcos(x)

Answer & Explanation

logik4z

logik4z

Expert

2022-01-25Added 8 answers

The function f(x)=cos2x has f(x+π)f(x), but any linear combination g of cos and sin has g(x+π)g(x)
Hana Larsen

Hana Larsen

Expert

2022-01-26Added 17 answers

Say we can do that, then
cos2xbcosx=asinx
Let t=cosx and square this equation. We get
t42bt3+b2t2=a2a2t2
which should be valid for all t[1,1] and there for for all t. So the polynomials must be equal for all t, so by comparing the coeficients we get 1=0. A contradiction.
So you can not express cos2x as linear combination of cosx  and  sinx

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