Given f(x)=Σ_(k=0) ^n ^n C_k sin(kx)cos((n−k)x) Find a simple form for f(x) (Your answer should be written like c(n).g(nx))


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Question Number 50908 by Smail last updated on 22/Dec/18

$G i v e n f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} s i n (k x) c o s ((n - k) x)$ $F i n d a s i m p l e f o r m f o r f (x)$ $(Y o u r a n s w e r s h o u l d b e w r i t t e n l i k e c (n) . g (n x))$
	Answered by Smail last updated on 23/Dec/18

	$f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} s i n (k x) c o s ((n - k) x)$ $l e t l = n - k$ $f (x) = \underset{l = 0}{\sum^{n}}^{n} C_{n - l} s i n ((n - l) x) c o s (l x)$ $= \underset{l = 0}{\sum^{n}}^{n} C_{l} s i n ((n - l) x) c o s (l x)$ $2 f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} (s i n ((n - k) x) c o s (k x) + s i n (k x) c o s ((n - k) x))$ $= \underset{k = 0}{\sum^{n}}^{n} C_{k} (s i n ((n - k) x + k x))$ $2 f (x) = \underset{k = 0}{\sum^{n}}^{n} C_{k} s i n (n x) = 2^{n} s i n (n x)$ $f (x) = 2^{n - 1} s i n (n x)$
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