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Prove-without-using-Mathematical-Induction-that-cos-n-x-1-2-n-1-k-0-n-1-2-C-k-n-cos-n-2k-x-where-x-is-any-real-number-and-n-is-any-positive-odd-integer-




Question Number 169295 by thfchristopher last updated on 28/Apr/22
Prove without using Mathematical   Induction that:  cos^n x=(1/2^(n−1) )Σ_(k=0) ^((n−1)/2) C_k ^n cos (n−2k)x   where  x is any real number and n is any  positive odd integer.
$$\mathrm{Prove}\:\mathrm{without}\:\mathrm{using}\:\mathrm{Mathematical}\: \\ $$$$\mathrm{Induction}\:\mathrm{that}: \\ $$$$\mathrm{cos}^{{n}} {x}=\frac{\mathrm{1}}{\mathrm{2}^{{n}−\mathrm{1}} }\underset{{k}=\mathrm{0}} {\overset{\left({n}−\mathrm{1}\right)/\mathrm{2}} {\sum}}{C}_{{k}} ^{{n}} \mathrm{cos}\:\left({n}−\mathrm{2}{k}\right){x}\: \\ $$$$\mathrm{where}\:\:{x}\:\mathrm{is}\:\mathrm{any}\:\mathrm{real}\:\mathrm{number}\:\mathrm{and}\:{n}\:\mathrm{is}\:\mathrm{any} \\ $$$$\mathrm{positive}\:\mathrm{odd}\:\mathrm{integer}. \\ $$

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