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http://functions.wolfram.com/01.07.23.0004.01
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Sum[(-1)^k Cos[a k + z], {k, 0, n}] == Cos[(1/2) (a + n (a + Pi))] Sec[a/2]
Cos[(1/2) n (a + Pi) + z]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sec </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> n </ci> <apply> <plus /> <ci> a </ci> <pi /> </apply> </apply> </apply> </apply> </apply> <apply> <sec /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> <apply> <plus /> <ci> a </ci> <pi /> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "n_"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k_"], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["a_", " ", "k_"]], "+", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["n", " ", RowBox[List["(", RowBox[List["a", "+", "\[Pi]"]], ")"]]]]]], ")"]]]], "]"]], " ", RowBox[List["Sec", "[", FractionBox["a", "2"], "]"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "n", " ", RowBox[List["(", RowBox[List["a", "+", "\[Pi]"]], ")"]]]], "+", "z"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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