If 8sin 3 -sin 3- - k-0 n a k cos k- is an identity is - where a k -s are constant- then n-

We have nbsp;∑ _ k=0 ^ n a _ k cos kθ nbsp; nbsp; =8sin ^ 3 θ nbsp;sin 3θ nbsp; nbsp;=8( 3sinθ nbsp; sin 3θ nbsp; nbsp;/ 4 nbsp;) ...

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Definition of Functions

If 8sin 3 θ...

Question

If $8 {sin}^{3} θ sin 3 θ = \sum_{k = 0}^{n} a_{k} cos k θ$ is an identity is $θ$ , where $a_{k}^{'} s$ are constant, then $n =$

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{sin}^{3} x sin 3 x = n \sum m = 0 c_{m} cos m x

x

c_{0}, c_{1}, . . . . . c_{n}

c_{n} \neq 0

n

t_{n} = \sum_{r = 0}^{n} \frac{1}{{({^{n} C}_{r})}^{k}}

S_{n} = \sum_{r = 0}^{n} \frac{r}{{({^{n} C}_{r})}^{k}}

k \in Z^{+}

{cos}^{- 1} (\frac{S_{n}}{n t_{n}})

{sin}^{3} x \cdot sin 3 x = n \sum m = 0 c_{m} \cdot cos m x

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S_{1} = \sum_{k = 0}^{n} {^{n} C}_{k} cos (k x) cos (n - k) x

S_{2} = \sum_{k = 0}^{n} {^{n} C}_{k} sin (k x) sin (n - k) x

S_{1} - S_{2}

{(1 - x^{3})}^{n} = \sum_{r = 0}^{n} a_{r} x^{r} {(1 - x)}^{3 n - 2 r}

a_{r}

n \in N

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