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Find constants a, b, c, d and e such that cos4x=a(sin^4)x+b(sin^3)x+c(sin^2)x+d(sin)x+e for all angles x. In other words, write cos(4x) as a polynomial in sin(x).

 Feb 27, 2024
 #1
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We know:

\(\cos(2x)=1-2 {\sin}^{2}(x)=2{\cos}^{2}(x)-1\).

Using a clever implementation of these formulas,

\(\cos(4x)=2{\cos}^{2}(2x)-1=2{(1-2{\sin}^{2}(x))}^{2}-1\).

We get: \(\cos(4x)=8{\sin}^{4}(x)-8{\sin}^{2}(x)+1\)

 Feb 27, 2024

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