hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = \map \sin {x + \frac {3 \pi} 2}       | r = \sin x \cos \frac {3 \pi} 2 + \cos x \sin \frac {3 \pi} 2       | c = Sine of Sum }} {{eqn | r = \sin x \cdot 0 + \cos x \cdot \paren {-1}       | c = Cosine of Three Right Angles and Sine of Three Right Angles }} {{eqn | r = -\cos x       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cos {x + \frac {3 \pi} 2}       | r = \cos x \cos \frac {3 \pi} 2 - \sin x \sin \frac {3 \pi} 2       | c = Cosine of Sum }} {{eqn | r = \cos x \cdot 0 - \sin x \cdot \paren {-1}       | c = Cosine of Three Right Angles and Sine of Three Right Angles }} {{eqn | r = \sin x       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \tan {x + \frac {3 \pi} 2}       | r = \frac {\map \sin {x + \frac {3 \pi} 2} } {\map \cos {x + \frac {3 \pi} 2} }        | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {-\cos x} {\sin x}       | c = Sine of Angle plus Three Right Angles and Cosine of Angle plus Three Right Angles }} {{eqn | r = -\cot x       | c = Cotangent is Cosine divided by Sine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cot {x + \frac {3 \pi} 2}       | r = \frac {\map \cos {x + \frac {3 \pi} 2} } {\map \sin {x + \frac {3 \pi} 2} }       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac {\sin x} {-\cos x}       | c = Cosine of Angle plus Right Angle and Sine of Angle plus Right Angle }} {{eqn | r = -\tan x       | c = Tangent is Sine divided by Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \sec {x + \frac {3 \pi} 2}       | r = \frac 1 {\map \cos {x + \frac {3 \pi} 2} }       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \frac 1 {\sin x}       | c = Cosine of Angle plus Three Right Angles }} {{eqn | r = \csc x       | c = Cosecant is Reciprocal of Sine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \csc {x + \frac {3 \pi} 2}       | r = \frac 1 {\map \sin {x + \frac {3 \pi} 2} }       | c = Cosecant is Reciprocal of Sine }} {{eqn | r = \frac 1 {-\cos x}       | c = Sine of Angle plus Three Right Angles }} {{eqn | r = -\sec x       | c = Secant is Reciprocal of Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \sin {\frac {3 \pi} 2 - \theta}       | r = \sin \frac {3 \pi} 2 \cos \theta - \cos \frac {3 \pi} 2 \sin \theta       | c = Sine of Difference }} {{eqn | r = \paren {-1} \times \cos \theta - 0 \times \sin \theta       | c = Sine of Three Right Angles and Cosine of Three Right Angles }} {{eqn | r = -\cos \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l=\cos \left({\frac {3 \pi} 2 - \theta}\right)       | r=\cos \frac {3 \pi} 2 \cos \theta + \sin \frac {3 \pi} 2 \sin \theta       | c=Cosine of Difference }} {{eqn | r=0 \times \cos \theta + \left({-1}\right) \times \sin \theta       | c=Cosine of Three Right Angles and Sine of Three Right Angles }} {{eqn | r=- \sin \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \sec {\frac {3 \pi} 2 - \theta}       | r = \frac 1 {\map \cos {\frac {3 \pi} 2 - \theta} }       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \frac 1 {-\sin \theta}       | c = Cosine of Three Right Angles less Angle }} {{eqn | r = -\csc \theta       | c = Cosecant is Reciprocal of Sine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \csc {\frac {3 \pi} 2 - \theta}       | r = \frac 1 {\map \sin {\frac {3 \pi} 2 - \theta} }       | c = Cosecant is Reciprocal of Sine }} {{eqn | r = \frac 1 {-\cos \theta}       | c = Sine of Three Right Angles less Angle }} {{eqn | r = -\sec \theta       | c = Secant is Reciprocal of Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 2 \theta + i \sin 2 \theta       | r = \paren {\cos \theta + i \sin \theta}^2       | c = De Moivre's Formula }} {{eqn | r = \cos^2 \theta + i^2 \sin^2 \theta + 2 i \cos \theta \sin \theta }} {{eqn | r = \cos^2 \theta - \sin^2 \theta + 2 i \cos \theta \sin \theta }} {{eqn | ll= \leadsto       | l = \sin 2 \theta       | r = 2 \cos \theta \sin \theta       | c = equating imaginary parts }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 2 \theta       | r = \map \sin {\theta + \theta} }} {{eqn | r = \sin \theta \cos \theta + \cos \theta \sin \theta       | c = Sine of Sum }} {{eqn | r = 2 \sin \theta \cos \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 2 \theta + i \sin 2 \theta       | r = \paren {\cos \theta + i \sin \theta}^2       | c = De Moivre's Formula }} {{eqn | r = \cos^2 \theta + i^2 \sin^2 \theta + 2 i \cos \theta \sin \theta }} {{eqn | r = \cos^2 \theta - \sin^2 \theta + 2 i \cos \theta \sin \theta }} {{eqn | ll= \leadsto       | l = \cos 2 \theta       | r = \cos^2 \theta - \sin^2 \theta       | c = equating real parts }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 2 \theta       | r = \map \cos {\theta + \theta} }} {{eqn | r = \cos \theta \cos \theta - \sin \theta \sin \theta       | c = Cosine of Sum }} {{eqn | r = \cos^2 \theta - \sin^2 \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \tan {2 \theta}       | r = \frac {\map \sin {2 \theta} } {\map \cos {2 \theta} }       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {2 \cos \theta \sin \theta} {\cos^2 \theta - \sin^2 \theta}       | c = Double Angle Formula for Sine and Double Angle Formula for Cosine }} {{eqn | r = \frac {\frac {2 \cos \theta \sin \theta} {\cos^2 \theta} } {\frac {\cos^2 \theta - \sin^2 \theta} {\cos^2 \theta} }       | c = dividing top and bottom by $\cos^2 \theta$ }} {{eqn | r = \frac {2 \tan \theta} {1 - \tan^2 \theta}       | c = Simplifying: Tangent is Sine divided by Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \tan {2 \theta}       | r = \map \tan {\theta + \theta} }} {{eqn | r = \frac {\tan \theta + \tan \theta} {1 - \tan \theta \tan \theta}       | c = Tangent of Sum }} {{eqn | r = \frac {2 \tan \theta} {1 - \tan^2 \theta} }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 4 \theta       | r = \frac {\sin 4 \theta} {\cos 4 \theta}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta} {8 \cos^4 \theta - 8 \cos^2 \theta + 1}       | c = Quadruple Angle Formula for Sine and Quadruple Angle Formula for Cosine }} {{eqn | r = \frac {4 \frac {\tan \theta} {\cos^2 \theta} - 8 \tan^3 \theta} {8 - \frac 8 {\cos^2 \theta} + \frac 1 {\cos^4 \theta} }       | c = dividing top and bottom by $\cos^4 \theta$ }} {{eqn | r =  \frac {4 \tan \theta \sec^2 \theta - 8 \tan^3 \theta} {8 - 8 \sec^2 \theta + \sec^4 \theta}       | c = Secant is Reciprocal of Cosine }} {{eqn | r =  \frac {4 \tan \theta \paren {1 + \tan^2 \theta} - 8 \tan^3 \theta} {8 - 8 \paren {1 + \tan^2 \theta} + \paren {1 + \tan^2 \theta}^2}       | c = Difference of Squares of Secant and Tangent }} {{eqn | r =  \frac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}       | c = multiplying out and gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 5 \theta       | r = \frac {\sin 5 \theta} {\cos 5 \theta}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta} {16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta}       | c = Quintuple Angle Formulas }} {{eqn | r = \frac {5 \frac {\tan \theta} {\cos^4 \theta} - 20 \frac {\tan^3 \theta} {\cos^2 \theta} + 16 \tan^5 \theta} {16 \cos \theta - \frac {20} {\cos^2 \theta} + \frac 5 {\cos^4 \theta} }       | c = dividing top and bottom by $\cos^5 \theta$ }} {{eqn | r = \frac {5 \tan \theta \sec^4 \theta - 20 \tan^3 \theta \sec^2 \theta + 16 \tan^5 \theta} {16 - 20 \sec^2 \theta + 5 \sec^4 \theta}       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \frac {5 \tan \theta \paren {1 + \tan^2 \theta}^2 - 20 \tan^3 \theta \paren {1 + \tan^2 \theta} + 16 \tan^5 \theta} {16 - 20 \paren {1 + \tan^2 \theta} + 5 \paren {1 + \tan^2 \theta}^2}       | c = Difference of Squares of Secant and Tangent }} {{eqn | r =  \frac {\tan^5 \theta - 10 \tan^3 \theta + 5 \tan \theta} {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}       | c = multiplying out and gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = 1 - 2 \sin^2 x       | r = \cos 2 x       | c = Double Angle Formula for Cosine: Corollary 2 }} {{eqn | ll= \leadsto       | l = \sin^2 x       | r = \frac {\cos 2 x - 1} {-2}       | c = solving for $\sin^2x$ }} {{eqn | r = \frac {1 - \cos 2 x} 2       | c = multiplying top and bottom by $-1$ and rearranging terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan^2 x       | r = \frac {\sin^2 x} {\cos^2 x}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\frac {1 - \cos 2 x} 2} {\frac {\cos 2 x + 1} 2}       | c = Square of Sine and Square of Cosine }} {{eqn | r = \frac {1 - \cos 2 x} {1 + \cos 2 x}       | c = multiplying top and bottom by $2$ }} {{end-eqn}} {{qed}} Category:Tangent Function ct8rk66qffzsan69mylvbe9ebjskfb7
{{begin-eqn}} {{eqn | l = \cos 3 x       | r = 4 \cos^3 x - 3 \cos x       | c = Triple Angle Formula for Cosine }} {{eqn | ll= \leadsto       | l = 4 \cos^3 x       | r = 3 \cos x + \cos 3 x       | c = rearranging }} {{eqn | ll= \leadsto       | l = \cos^3 x       | r = \dfrac {3 \cos x + \cos 3 x} 4       | c = dividing both sides by $4$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin^4 x       | r = \paren {\sin^2 x}^2 }} {{eqn | r = \paren {\frac {1 - \cos 2 x} 2}^2       | c = Square of Sine }} {{eqn | r = \frac {1 - 2 \cos 2 x + \cos^2 2 x} 4       | c = multiplying out }} {{eqn | r = \frac {1 - 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4       | c = Square of Cosine }} {{eqn | r = \frac {2 - 4 \cos 2 x + 1 + \cos 4 x} 8       | c = multiplying top and bottom by $2$ }} {{eqn | r = \frac {3 - 4 \cos 2 x + \cos 4 x} 8       | c = rearrangement }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 5 x       | r = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x       | c = Quintuple Angle Formula for Sine }} {{eqn | ll= \leadsto       | l = 16 \sin^5 x       | r = \sin 5 x + 20 \sin^3 x - 5 \sin x       | c = rearranging }} {{eqn | r = \sin 5 x + 20 \paren {\frac {3 \sin x - \sin 3 x} 4} - 5 \sin x       | c = Power Reduction Formula for Cube of Sine }} {{eqn | r = \sin 5 x + 15 \sin x - 5 \sin 3 x - 5 \sin x       | c = multipying out }} {{eqn | r = 10 \sin x - 5 \sin 3 x + \sin 5 x       | c = rearranging }} {{eqn | ll= \leadsto       | l = \sin^5 x       | r = \frac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}       | c = dividing both sides by 16 }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 5 x       | r = 16 \cos^5 x - 20 \cos^3 x + 5 \cos x       | c = Quintuple Angle Formula for Cosine }} {{eqn | ll= \leadsto       | l = 16 \cos^5 x       | r = \cos 5 x + 20 \cos^3 x - 5 \cos x       | c = rearranging }} {{eqn | r = \cos 5 x + 20 \paren {\frac {3 \cos x + \cos 3 x} 4} - 5 \cos x       | c = Power Reduction Formula for Cube of Sine }} {{eqn | r = \cos 5 x + 15 \cos x + 5 \cos 3 x - 5 \cos x       | c = multipying out }} {{eqn | r = 10 \cos x + 5 \cos 3 x + \cos 5 x       | c = rearranging }} {{eqn | ll= \leadsto       | l = \cos^5 x       | r = \frac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}       | c = dividing both sides by 16 }} {{end-eqn}} {{qed}}
Let $B \subseteq A^\perp$. Then by Orthocomplement Reverses Subset: :$A^{\perp\perp} \subseteq B^\perp$ By Double Orthocomplement is Closed Linear Span and the definition of closed linear span: :$A \subseteq A^{\perp\perp}$ Hence, by Subset Relation is Transitive: :$A \subseteq B^\perp$ {{qed}} Category:Hilbert Spaces llqojzv5zg3rru2iy6w3z2fl6ud7645
{{begin-eqn}} {{eqn | o =        | r = \frac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2 }} {{eqn | r = \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} + \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2       | c = Cosine of Difference and Cosine of Sum }} {{eqn | r = \frac {2 \cos \alpha \cos \beta} 2 }} {{eqn | r = \cos \alpha \cos \beta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =       | r = \frac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2 }} {{eqn | r = \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} - \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2       | c = Cosine of Difference and Cosine of Sum }} {{eqn | r = \frac {2 \sin \alpha \sin \beta} 2 }} {{eqn | r = \sin \alpha \sin \beta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =        | r = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}       | c =  }} {{eqn | r = 2 \frac {\map \sin {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} + \map \sin {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} } 2       | c = Simpson's Formula for Sine by Cosine }} {{eqn | r = \sin \frac {2 \alpha} 2 + \sin \frac {2 \beta} 2       | c =  }} {{eqn | r = \sin \alpha + \sin \beta       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =        | r = 2 \map \cos {\frac {\alpha + \beta} 2} \map \sin {\frac {\alpha - \beta} 2}       | c =  }} {{eqn | r = 2 \frac {\map \sin {\dfrac {\alpha - \beta} 2 + \dfrac {\alpha + \beta} 2} + \map \sin {\dfrac {\alpha - \beta} 2 - \dfrac {\alpha + \beta} 2} } 2       | c = Simpson's Formula for Sine by Cosine }} {{eqn | r = \sin \frac {2 \alpha} 2 - \sin \frac {2 \beta} 2       | c =  }} {{eqn | r = \sin \alpha - \sin \beta       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =        | r = 2 \, \map \cos {\frac {\alpha + \beta} 2} \, \map \cos {\frac {\alpha - \beta} 2}       | c =  }} {{eqn | r = 2 \frac {\map \cos {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} + \map \cos {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} } 2       | c = Simpson's Formula for Cosine by Cosine }} {{eqn | r = \cos \frac {2 \beta} 2 + \cos \frac {2 \alpha} 2       | c =  }} {{eqn | r = \cos \alpha + \cos \beta       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =        | r = -2 \, \map \sin {\frac {\alpha + \beta} 2} \, \map \sin {\frac {\alpha - \beta} 2}       | c =  }} {{eqn | r = -2 \frac {\map \cos {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} - \map \cos {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} } 2       | c = Simpson's Formula for Sine by Sine }} {{eqn | r = -\paren {\cos \frac {2 \beta} 2 - \cos \frac {2 \alpha} 2}       | c =  }} {{eqn | r = \cos \alpha - \cos \beta       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =        | r = \frac {\sin \left({\alpha + \beta}\right) - \sin \left({\alpha - \beta}\right)} 2 }} {{eqn | r = \frac {\left({\sin \alpha \cos \beta + \cos \alpha \sin \beta}\right) - \left({\sin \alpha \cos \beta - \cos \alpha \sin \beta}\right)} 2       | c = Sine of Sum and Sine of Difference }} {{eqn | r = \frac {2 \cos \alpha \sin \beta} 2 }} {{eqn | r = \cos \alpha \sin \beta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos \alpha \sin \beta       | r = \sin \beta \cos \alpha }} {{eqn | r = \frac {\sin \paren {\beta + \alpha} + \sin \paren {\beta - \alpha} } 2       | c = Simpson's Formula for Sine by Cosine }} {{eqn | r = \frac {\sin \paren {\alpha + \beta} + \sin \paren {-\paren {\alpha - \beta} } } 2       | c =  }} {{eqn | r = \frac {\sin \paren {\alpha + \beta} - \sin \paren {\alpha - \beta} } 2       | c = Sine Function is Odd }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin^{2 n + 1} \theta       | r = \paren {\frac {e^{i \theta} - e^{-i \theta} } {2 i} }^{2 n + 1}       | c = Sine Exponential Formulation }} {{eqn | r = \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i}^{2 n + 1} }       | c = Power of Product }} {{eqn | r = \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i} \paren {2}^{2 n} \paren {i}^{2 n} }       | c =  }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1}       | c = $i^2 = -1$, Exponent Combination Laws/Power of Power }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{\paren {2 n + 1 - k } i \theta} \paren {-1}^{k} e^{-\paren {k} i \theta}       | c = Binomial Theorem }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \sum^{2 n + 1}_{k \mathop = 0} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k } i \theta}       | c = Exponential of Sum }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k } i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k } i \theta} }       | c = partitioning the sum }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \paren {-1}^{2 n + 1 - k} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 n + 1 - 2 \paren {2 n + 1 - k } } i \theta} }       | c = $k \mapsto 2 n + 1 - k$ }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \paren {-1}^{2 n} \paren {-1}^{1} \paren {-1}^{- k} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 n + 1 - 2 \paren {2 n + 1 - k } } i \theta} }       | c = Exponent Combination Laws/Product of Powers }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} - \sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} }       | c = Symmetry Rule for Binomial Coefficients }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k \frac {e^{\paren {2 n - 2 k + 1} i \theta} - e^{-\paren {2 n - 2 k + 1} i \theta} } {2i} }       | c = }} {{eqn | r = \frac {\paren {-1}^n} {2^{2 n} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom {2 n + 1} k \sin \paren {2 n - 2 k + 1} \theta       | c = Sine Exponential Formulation }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos^{2 n + 1} \theta       | r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^{2 n + 1}       | c = Cosine Exponential Formulation }} {{eqn | r = \frac 1 {2^{2 n + 1} } \paren {e^{i \theta} + e^{-i \theta} }^{2 n + 1}       | c = Exponent Combination Laws: Power of Product }} {{eqn | r = \frac 1 {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{k i \theta} e^{-\paren {2 n - k + 1} i \theta}       | c = Binomial Theorem }} {{eqn | r = \frac 1 {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta}       | c = Exponential of Sum }} {{eqn | r = \frac 1 {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} }       | c = Partitioning the sum }} {{eqn | r = \frac 1 {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 \paren {2 n - k + 1} - 2 n - 1} i \theta} }       | c = $k \mapsto 2 n + 1 - k$ }} {{eqn | r = \frac 1 {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} }       | c = Symmetry Rule for Binomial Coefficients }} {{eqn | r = \frac 1 {2^{2 n} } \sum^n_{k \mathop = 0} \binom {2 n + 1} k \, \map \cos {2 n - 2 k + 1} \theta       | c = Cosine Exponential Formulation }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos^n \theta       | r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n       | c = De Moivre's Theorem }} {{eqn | r = \frac {\paren {e^{i \theta} + e^{-i \theta} }^n} {2^n}       | c =  }} {{eqn | r = \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - k} i \theta} e^{-k i \theta}       | c = Binomial Theorem }} {{eqn | r = \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - 2 k} i \theta}       | c =  }} {{end-eqn}} Matching up terms from the beginning of this expansion with those from the end: {{begin-eqn}} {{eqn | l = 2^n \cos^n \theta       | r = e^{n i \theta} + \binom n 1 e^{\paren {n - 2} i \theta} + \binom n 2 e^{\paren {n - 4} i \theta} + \cdots       | c =  }} {{eqn | o =       | ro= +       | r = \binom n {n - 2} e^{-\paren {n - 4} i \theta} + \binom n {n - 1} e^{-\paren {n - 2} i \theta} + e^{-n i \theta}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {2 n} 2 \cos^n \theta       | r = \paren {\frac {e^{n i \theta} + e^{-n i \theta} } 2} + \binom n 1 \paren {\frac {e^{\paren {n - 2} i \theta} + e^{-\paren {n - 2} i \theta} } 2}       | c =  }} {{eqn | o =       | ro= +       | r = \binom n 2 \paren {\frac {e^{\paren {n - 4} i \theta} + e^{-\paren {n - 4} i \theta} } 2} + \cdots       | c =  }} {{end-eqn}} Thus: :$\cos^n \theta = \dfrac 1 {2^{n - 1} } \paren {\cos n \theta + n \cos \paren {n - 2} \theta + \dfrac {n \paren {n - 1} } {2!} \cos \paren {n - 4} \theta + \cdots + R_n}$ Now to determine $R_n$. The middle two terms of the sequence $0, 1, \ldots, n$ are $\dfrac {n - 1} 2$ and $\dfrac {n + 1} 2$. Thus, when $k = \dfrac {n - 1} 2$: {{begin-eqn}} {{eqn | l = n - 2 k       | r = n - 2 \frac {n - 1} 2       | c =  }} {{eqn | r = n - \frac {2 n - 2} 2       | c =  }} {{eqn | r = n - n + 1       | c =  }} {{eqn | r = 1       | c =  }} {{end-eqn}} Similarly, when $k = \dfrac {n + 1} 2$: {{begin-eqn}} {{eqn | l = n + 2 k       | r = n - 2 \frac {n + 1} 2       | c =  }} {{eqn | r = n - \frac {2 n + 2} 2       | c =  }} {{eqn | r = n - n - 1       | c =  }} {{eqn | r = -1       | c =  }} {{end-eqn}} The binomial coefficient in each case is the same, because: {{begin-eqn}} {{eqn | l = n - \frac {n - 1} 2       | r = \frac {2 n - n - 1} 2       | c =  }} {{eqn | r = \frac {n + 1} 2       | c =  }} {{end-eqn}} So: {{begin-eqn}} {{eqn | l = \binom n {\paren {n - 1} / 2}       | r = \frac {n!} {\paren {\frac {n - 1} 2}! \paren {\frac {n + 1} 2}!}       | c =  }} {{eqn | r = \binom n {\paren {n + 1} / 2}       | c =  }} {{end-eqn}} Thus the two middle terms collapse to: {{begin-eqn}} {{eqn | l = R_n       | r = \frac {n!} {\paren {\frac {n - 1} 2}! \paren {\frac {n + 1} 2}!} \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}       | c =  }} {{eqn | r = \frac {n!} {\paren {\frac {n - 1} 2}! \paren {\frac {n + 1} 2}!} \cos \theta       | c =  }} {{end-eqn}} {{qed}}