poveli1e

2022-01-24

Write ${\mathrm{cos}}^{2}\left(x\right)$ as linear combination of $x↦\mathrm{sin}\left(x\right)$ and $x↦\mathrm{cos}\left(x\right)$

logik4z

Expert

The function $f\left(x\right)\phantom{\rule{0.222em}{0ex}}={\mathrm{cos}}^{2}x$ has $f\left(x+\pi \right)\equiv f\left(x\right)$, but any linear combination g of $\mathrm{cos}$ and $\mathrm{sin}$ has $g\left(x+\pi \right)\equiv -g\left(x\right)$

Hana Larsen

Expert

Say we can do that, then
${\mathrm{cos}}^{2}x-b\mathrm{cos}x=a\mathrm{sin}x$
Let $t=\mathrm{cos}x$ and square this equation. We get
${t}^{4}-2b{t}^{3}+{b}^{2}{t}^{2}={a}^{2}-{a}^{2}{t}^{2}$
which should be valid for all $t\in \left[-1,1\right]$ 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 ${\mathrm{cos}}^{2}x$ as linear combination of

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