Cosine of Integer Multiple of Argument/Formulation 7

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Theorem

For $n \in \Z_{>0}$:

\(\ds \cos n \theta\) \(=\) \(\ds \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - \paren {2 \sin \theta} \paren {\map \sin {\paren {n - 1} \theta} + \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} + \cdots}\)
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\)


Proof

The proof proceeds by induction.


For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:

$\ds \cos n \theta = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$


Basis for the Induction

$\map P 1$ is the case:

\(\ds \cos \theta\) \(=\) \(\ds \cos \theta\)
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {1 + 1} \pi} 2 + \paren {\sin^2 \frac \pi 2} \cos \theta - 2 \sin \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {1 - 0} \theta} }\)

So $\map P 1$ is seen to hold.


$\map P 2$ is the case:

\(\ds \cos 2 \theta\) \(=\) \(\ds 1 - 2 \sin^2 \theta\) Double Angle Formula for Cosine: Corollary $2$
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {2 + 1} \pi} 2 + \paren {\sin^2 \frac {2 \pi} 2} \cos \theta - 2 \sin \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {2 - 1} \theta} }\)

So $\map P 2$ is also seen to hold.


$\map P 3$ is the case:

\(\ds \cos 3 \theta\) \(=\) \(\ds -2 \sin \theta \sin 2 \theta + \cos \theta\) Cosine of Integer Multiple of Argument: Formulation 6
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {3 + 1} \pi} 2 + \paren {\sin^2 \frac {3 \pi} 2} \cos \theta - 2 \sin \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {3 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {3 - 1} \theta} + \paren {\sin \frac {2 \pi} 2} \map \sin {\paren {3 - 2} \theta} }\)

So $\map P 3$ 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 $n > 3$ and $n$ even, then it logically follows that $\map P {n + 2}$ is true.

We also need to show that, if $\map P n$ is true, where $n > 3$ and $n$ odd, then it logically follows that $\map P {n + 2}$ is true.


So this is our induction hypothesis:

$\ds \map \cos {n \theta} = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$

from which we are to show:

$\ds \map \cos {\paren {n + 2} \theta} = \sin^2 \frac {\paren {n + 3} \pi} 2 + \paren {\sin^2 \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n + 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }$


Induction Step

This is our induction step:

For $n$ even or for $n$ odd: (Identical argument)

\(\ds \map \cos {\paren {n + 2} \theta}\) \(=\) \(\ds \paren {-2 \sin \theta} \map \sin {\paren {n + 1} \theta} + \map \cos {\paren n \theta}\) Cosine of Integer Multiple of Argument: Formulation 6
\(\ds \) \(=\) \(\ds \paren {-2 \sin \theta} \map \sin {\paren {n + 1} \theta} + \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\)
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - \paren {2 \sin \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {n + 2 - 0} \theta} + \paren {\sin \frac {\pi} 2} \map \sin {\paren {n + 2 - 1} \theta} } - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\) rearranging terms and adding zero
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {n + 3} \pi} 2 + \paren {\sin^2 \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} - \paren {2 \sin \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {n + 2 - 1} \theta} } - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\) Sine of Angle plus Integer Multiple of Pi
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {n + 3} \pi} 2 + \paren {\sin^2 \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} - \paren {2 \sin \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {n + 2 - 1} \theta} } - 2 \sin \theta \paren {\sum_{k \mathop = 2}^{n + 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds \sin^2 \frac {\paren {n + 3} \pi} 2 + \paren {\sin^2 \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n + 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }\)


The result follows by the Principle of Mathematical Induction.

Therefore:

$\ds \forall n \in \Z_{>0}: \cos n \theta = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$

$\blacksquare$


Examples

Cosine of Quintuple Angle

$\cos 5 \theta = \cos \theta - 2 \sin \theta \paren {\sin 4 \theta + \sin 2 \theta}$


Cosine of Sextuple Angle

$\map \cos {6 \theta } = 1 - 2 \sin \theta \paren {\sin 5 \theta + \sin 3 \theta + \sin \theta}$
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