+26367 Express \(\cos(5x)\) as a polynomial in \(\cos(x)\).
\(\begin{array}{|rcll|} \hline \mathbf{\Big(\cos(x) + i\sin(x) \Big)^5} &=& \mathbf{\cos(5x) +i\sin(5x)} \\\\ \Big(\cos(x) + i\sin(x) \Big)^5 &=& \binom50\cos^5(x) \\ &+& \binom51\cos^4(x)(i)\sin(x) \\ &+& \binom52\cos^3(x)(i^2)\sin^2(x) \quad & | \quad i^2=-1 \\ &+& \binom53\cos^2(x)(i^3)\sin^3(x) \quad & | \quad i^3=-i \\ &+& \binom54\cos(x)(i^4)\sin^4(x) \quad & | \quad i^4=1 \\ &+& \binom55(i^5)\sin^5(x) \quad & | \quad i^5=i \\\\ &=& \binom50\cos^5(x) \\ &+& \binom51\cos^4(x)(i)\sin(x) \\ &-& \binom52\cos^3(x)\sin^2(x) \\ &-& \binom53\cos^2(x)(i)\sin^3(x) \\ &+& \binom54\cos(x)\sin^4(x) \\ &+& \binom55(i)\sin^5(x) \\ \hline \end{array} \)
Compare the real parts of each side:
\(\begin{array}{|rcll|} \hline \cos(5x) &=& \binom50\cos^5(x) \\ &-& \binom52\cos^3(x)\sin^2(x) \\ &+& \binom54\cos(x)\sin^4(x) \\\\ \cos(5x) &=& \binom50\cos^5(x) \quad & | \quad \binom50 = 1 \\ &-& \binom52\cos^3(x)\sin^2(x) \quad & | \quad \binom52 = 10,\ \sin^2(x)=1-\cos^2(x) \\ &+& \binom54\cos(x)\sin^2(x)\sin^2(x) \quad & | \quad \binom54 = 5,\ \sin^2(x)=1-\cos^2(x) \\\\ \cos(5x) &=& \cos^5(x) \\ &-& 10\cos^3(x)\Big(1-\cos^2(x)\Big) \\ &+& 5\cos(x)\Big(1-\cos^2(x)\Big)\Big(1-\cos^2(x)\Big) \\\\ \cos(5x) &=& \cos^5(x) \\ &-& 10\cos^3(x) + 10\cos^5(x) \\ &+& 5\cos(x)\Big(1-\cos^2(x)\Big)^2 \\\\ \cos(5x) &=& \cos^5(x) \\ &-& 10\cos^3(x) + 10\cos^5(x) \\ &+& 5\cos(x)\Big(1-2\cos^2(x)+\cos^4(x)\Big) \\\\ \cos(5x) &=& \cos^5(x) \\ &-& 10\cos^3(x) + 10\cos^5(x) \\ &+& 5\cos(x) -10\cos^3(x)+5\cos^5(x) \\\\ \mathbf{\cos(5x)} &=& \mathbf{ 16\cos^5(x) - 20\cos^3(x) + 5\cos(x) } \\ \hline \end{array}\)
