The proof proceeds by induction. For all $n \in \Z_{>0}$, let $\map P n$ be the proposition: :$\displaystyle \cos n \theta = \dfrac 1 2 \paren {\paren {2 \cos \theta }^n + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac n k \dbinom {n - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{n - 2 k } }$ === Basis for the Induction === $\map P 1$ is the case: {{begin-eqn}} {{eqn | l = \cos \theta       | r = \cos \theta       | c =  }} {{eqn | r = \frac 1 2 \paren {2 \cos \theta}^1       | c =  }} {{end-eqn}} So $\map P 1$ is seen to hold. $\map P 2$ is the case: {{begin-eqn}} {{eqn | l = \cos 2 \theta       | r = 2 \cos^2 \theta - 1       | c = Double Angle Formulas/Cosine/Corollary 1 }} {{eqn | r = \dfrac 1 2 \paren {\paren {2 \cos \theta }^2 - \dfrac 2 1 \dbinom {2 - \paren {0 + 1 } } {1 - 1} \paren {2 \cos \theta }^{2 - 2 } }       | c =  }} {{end-eqn}} So $\map P 2$ is also seen to hold. This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\map P n$ is true, where $n > 2$, then it logically follows that $\map P {n + 1}$ is true. So this is our induction hypothesis: :$\displaystyle \map \cos {n \theta} = \dfrac 1 2 \paren {\paren {2 \cos \theta }^n + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac n k \dbinom {n - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{n - 2 k } }$ from which we are to show: :$\displaystyle \map \cos {\paren {n + 1} \theta} = \dfrac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac {n + 1 } k \dbinom {n + 1 - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{n + 1 - 2 k } }$ === Induction Step === This is our induction step: To proceed, we will require the following Lemma: === Lemma === {{:Sine of Integer Multiple of Argument/Formulation 1/Lemma}}{{qed|lemma}} Dividing through by $\sin \theta$, we obtain: {{begin-eqn}} {{eqn | l = \map \cos {n \theta}       | r = \frac {\map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta} } {\map \sin {\theta} } }} {{end-eqn}} We now have: {{begin-eqn}} {{eqn | l = \map \cos {\paren {n + 1} \theta}        | r = \map \cos {n \theta + \theta }       | c = }} {{eqn | r = \cos n \theta \cos \theta - \sin n \theta \sin \theta       | c = Cosine of Sum }} {{eqn | r = \paren {\frac {\map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta} } {\map \sin {\theta} } } \cos \theta - \sin n \theta \sin \theta       | c = substitution from above }} {{eqn | r = \sin n \theta \frac {\paren {\cos^2 \theta - \sin^2 \theta } } {\sin \theta } - \map \sin {\paren {n - 1 } \theta} \frac {\cos \theta } {\sin \theta }       | c = group terms }} {{eqn | r = \frac {\map \sin {n \theta} } {\sin \theta } \paren {2 \cos^2 \theta - 1 } - \frac {\map \sin {\paren {n - 1 } \theta} } {\sin \theta } \cos \theta       | c = Double Angle Formulas/Cosine/Corollary 1 }} {{eqn | r = \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } } \paren {2 \cos^2 \theta - 1 } - \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - 1 - \paren {k + 1} } k \paren {2 \cos \theta}^{n - 1 - \paren {2 k + 1} } } \cos \theta       | c = Sine of Integer Multiple of Argument/Formulation 1 }} {{eqn | r = \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } } \paren {\frac {\paren {2 \cos \theta }^2 } 2 - 1 } - \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - 1 - \paren {k + 1} } k \paren {2 \cos \theta}^{n - 1 - \paren {2 k + 1} } } \frac {\paren {2 \cos \theta } } 2       | c = multiply by $1$ }} {{eqn | r = \frac 1 2 \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } } - \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } } - \frac 1 2 \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 2} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }       | c =  }} {{eqn | r = \frac 1 2 \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } } - \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \paren {\binom {n - \paren {k + 1} } k + \frac 1 2 \binom {n - \paren {k + 2} } k } \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }       | c =  }} {{eqn | r = \frac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } } + \paren {\sum_{k \mathop \ge 1} \paren {-1}^k \paren {\binom {n - \paren {\paren {k - 1 } + 1} } {k - 1 } + \frac 1 2 \binom {n - \paren {\paren {k - 1 } + 2} } {k - 1 } } \paren {2 \cos \theta}^{n - \paren {2 \paren {k - 1 } + 1} } }       | c = start all sums at $k = 1$ }} {{eqn | r = \frac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1}^k \paren {\binom {n - \paren {k + 1} } k + 2 \binom {n - k } {k - 1} + \binom {n - \paren {k + 1} } {k - 1 } } \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } }       | c = grouping all terms in one sum }} {{eqn | r = \frac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1}^k \paren {\binom {n - k } k + \binom {n - k } {k - 1} + \binom {n - k } {k - 1 } } \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } }       | c = Pascal's Rule }} {{eqn | r = \frac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1}^k \paren {\binom {n + 1 - k } k + \binom {n - k } {k - 1 } } \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } }       | c = Pascal's Rule }} {{eqn | r = \frac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1}^k \paren {\paren {\frac {n + 1 - k } k} \binom {n - k } {k - 1 } + \binom {n - k } {k - 1 } } \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } }       | c = Factors of Binomial Coefficient }} {{eqn | r = \frac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1}^k \paren {\paren {\frac {n + 1 - k } k + 1 } \binom {n - k } {k - 1 } } \paren {2 \cos \theta}^{n + 1 - \paren {2 k } } }       | c =  }} {{eqn | r = \dfrac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac {n + 1 } k \binom {n - k } {k - 1} \paren {2 \cos \theta }^{n + 1 - 2 k } }       | c =  }} {{eqn | r = \dfrac 1 2 \paren {\paren {2 \cos \theta }^{n + 1 } + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac {n + 1 } k \binom {n + 1 - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{n + 1 - 2 k } }       | c = adding and subtracting $1$ }} {{end-eqn}} The result follows by the Principle of Mathematical Induction. Therefore: :$\displaystyle \forall n \in \Z_{>0}: \cos n \theta = \dfrac 1 2 \paren {\paren {2 \cos \theta }^n + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac n k \dbinom {n - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{n - 2 k } }$ {{qed}}