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anemone
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Express $\cos^7 x+\cos^7 \left( x+\dfrac{2 \pi}{3} \right)+\cos^7 \left( x+\dfrac{4 \pi}{3} \right)$ in terms of $\cos 3x$.
[sp]This is best done using complex numbers. Let $\lambda = e^{ix}$, $\omega = e^{2\pi i/3}$. Then $\cos x = \frac12(\lambda + \lambda^{-1})$, $\cos \left( x+\frac{2 \pi}{3} \right) = \frac12(\lambda\omega + \lambda^{-1}\omega^{-1})$, $\cos \left( x+\frac{4 \pi}{3} \right) = \frac12(\lambda\omega^2 + \lambda^{-1}\omega^{-2})$, and $$ \cos^7 x+\cos^7 \left( x+\tfrac{2 \pi}{3} \right)+\cos^7 \left( x+\tfrac{4 \pi}{3} \right) = 2^{-7}\bigl((\lambda + \lambda^{-1})^7 + (\lambda\omega + \lambda^{-1}\omega^{-1})^7 + (\lambda\omega^2 + \lambda^{-1}\omega^{-2})^7\bigr).$$ Now expand by the binomial theorem and use the facts that $\omega^3=1$ and $1+\omega+\omega^2=0$: $$\begin{aligned} \cos^7 x+\cos^7 \left( x+\tfrac{2 \pi}{3} \right)+\cos^7 \left( x+\tfrac{4 \pi}{3} \right) &= 2^{-7}\bigl(\lambda^7(1+\omega^7+\omega^{14}) + 7\lambda^5(1+\omega^5+\omega^{10}) + 21\lambda^3(1+\omega^3+\omega^6) + 28\lambda(1+\omega+\omega^2)\bigr. \\ & \qquad {} + \bigl.28\lambda^{-1}(1+\omega^{-1}+\omega^{-2}) + 21\lambda^{-3}(1+\omega^{-3}+\omega^{-6}) + 7\lambda^{-5}(1+\omega^{-5}+\omega^{-10}) + \lambda^{-7}(1+\omega^{-7}+\omega^{-14})\bigr) \\ &= 2^{-7}\bigl((\lambda^{7} + \lambda^{-7})(1+\omega+\omega^2) + 7(\lambda^{5} + \lambda^{-5})(1+\omega+\omega^2) \\ & \qquad {} + 21(\lambda^{3} + \lambda^{-3})(1+1+1) + 28(\lambda + \lambda^{-1})(1+\omega+\omega^2)\bigr) \\ &= 2^{-7}\cdot21\cdot3(\lambda^{3} + \lambda^{-3}) = \frac{63}{2^6}\cos(3x). \end{aligned}$$ Thus the answer is $(1-2^{-6})\cos(3x).$[/sp]anemone said:Express $\cos^7 x+\cos^7 \left( x+\dfrac{2 \pi}{3} \right)+\cos^7 \left( x+\dfrac{4 \pi}{3} \right)$ in terms of $\cos 3x$.
anemone said:Thank you again Opalg for participating! I think we used the pretty same approach, because the coefficients of the terms that we have in our methods are all the same.
My solution:
According to the power reduction formula, we have
$\cos^7 x=\dfrac{35\cos x+21\cos 3x+7\cos5x+\cos7x}{64}$
hence
$\cos^7 \left(x+\dfrac{2 \pi}{3} \right)=\dfrac{35\cos \left(x+\dfrac{2 \pi}{3} \right)+21\cos 3\left(x+\dfrac{2 \pi}{3} \right)+7\cos5\left(x+\dfrac{2 \pi}{3} \right)+\cos7\left(x+\dfrac{2 \pi}{3} \right)}{64}$
$\cos^7 \left(x+\dfrac{4 \pi}{3} \right)=\dfrac{35\cos \left(x+\dfrac{4 \pi}{3} \right)+21\cos 3\left(x+\dfrac{4 \pi}{3} \right)+7\cos5\left(x+\dfrac{4 \pi}{3} \right)+\cos7\left(x+\dfrac{4 \pi}{3} \right)}{64}$
Notice that
$35\cos x+35\cos \left(x+\dfrac{2 \pi}{3} \right)+35\cos \left(x+\dfrac{4 \pi}{3} \right)=35\left( \cos x+\cos \left(x+\dfrac{2 \pi}{3} \right) \right)+35\cos \left(x+\dfrac{4 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=35\left( 2 \cos \left(\dfrac{ \pi}{3} \right) \cos \left( x+\dfrac{ \pi}{3} \right) \right)+35\cos \left(x+\dfrac{4 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=35 \cos \left( x+\dfrac{ \pi}{3} \right) +35\cos \left(x+\dfrac{4 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=35 \left( \cos \left( x+\dfrac{ \pi}{3} \right) +\cos \left(x+\dfrac{4 \pi}{3} \right) \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=35 \left( 2\cos \left( \dfrac{ \pi}{2} \right) \cos \left(x+\dfrac{5 \pi}{6} \right) \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0$
Similarly,
$21\cos 3x+21\cos 3\left(x+\dfrac{2 \pi}{3} \right)+21\cos 3\left(x+\dfrac{4 \pi}{3} \right)=21\cos 3x+21\cos (3x+2 \pi)+21\cos (3x+2 \pi)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=21\cos 3x+21\cos 3x+21\cos 3x$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=63\cos 3x$
$7\cos5x+7\cos5\left(x+\dfrac{2 \pi}{3} \right)+7\cos5\left(x+\dfrac{4 \pi}{3} \right)=7\left( \cos 5x+\cos 5\left(x+\dfrac{2 \pi}{3} \right) \right)+7\cos 5\left(x+\dfrac{4 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=7\left( 2 \cos \left(\dfrac{ 5\pi}{3} \right) \cos \left( 5x+\dfrac{ 5\pi}{3} \right) \right)+7\cos \left(5x+\dfrac{20 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=7 \cos \left( 5x+\dfrac{ 5\pi}{3} \right) +7\cos \left(5x+\dfrac{20 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=7 \left( \cos \left( 5x+\dfrac{ 5\pi}{3} \right) +\cos \left(5x+\dfrac{20 \pi}{3} \right) \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=7 \left( 2\cos \left( \dfrac{ 5\pi}{2} \right) \cos \left(5x+\dfrac{25 \pi}{6} \right) \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0$
and finally
$\cos7x+\cos7\left(x+\dfrac{2 \pi}{3} \right)+\cos7\left(x+\dfrac{4 \pi}{3} \right)=\left( \cos 7x+\cos 7\left(x+\dfrac{2 \pi}{3} \right) \right)+\cos 7\left(x+\dfrac{4 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left( 2 \cos \left(\dfrac{ 7\pi}{3} \right) \cos \left( 7x+\dfrac{ 7\pi}{3} \right) \right)+\cos \left(7x+\dfrac{28 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \cos \left( 5x+\dfrac{ 5\pi}{3} \right) +7\cos \left(5x+\dfrac{20 \pi}{3} \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left( \cos \left( 7x+\dfrac{ 7\pi}{3} \right) +\cos \left(7x+\dfrac{28 \pi}{3} \right) \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=2 \left( 2\cos \left( \dfrac{ 21\pi}{2} \right) \cos \left(7x+\dfrac{35 \pi}{6} \right) \right)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0$
Therefore
$\cos^7 x+\cos^7 \left( x+\dfrac{2 \pi}{3} \right)+\cos^7 \left( x+\dfrac{4 \pi}{3} \right)=0+\dfrac{63 \cos 3x}{64}+0+0=\dfrac{63 \cos 3x}{64}$
kaliprasad said:1st step to decompose $cos ^7x$ as sum of cos x , cos 3x terms etc is good
using cos x + cos $( x+\dfrac{2 \pi}{3}$ ) + cos $( x+\dfrac{4 \pi}{3} $) = 0 for 5x and 7x can be made as zero
as cos 5 $( x+\dfrac{2 \pi}{3} $) = cos $( 5 x+\dfrac{4 \pi}{3} $) so on and all terms except 3x can be made zero. this reduces number of steps
anemone said:I should have thought of this earlier so that that would save me all the trouble struggling with the latex!
\begin{align}...\end{align}
mathbalarka said:One way of aligning the equation is to use
to save your time.Code:\begin{align}...\end{align}
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