Cosine of Integer Multiple of Argument/Formulation 8

Theorem
For $n \in \Z_{>1}$:


 * $\cos n \theta = \map \cos {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-2} - \cfrac 1 {a_{n-1}}} }}} }$

where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$ and
 * A terminal $a_{n-1} = \cos \theta$ term.

Proof
From Cosine of Integer Multiple of Argument Formulation 4 we have:

Therefore $a_0 = 2 \cos \theta$

Once again, from Cosine of Integer Multiple of Argument Formulation 4 we have:

In the equations above, let $n = n - k$:

Therefore $a_1 = a_2 = \cdots = a_{n-2} = 2 \cos \theta$

Finally, let $k = n - 2$, then:

Therefore $a_{n - 1} = \cos \theta$

Therefore:

For $n \in \Z_{>1}$:


 * $\cos n \theta = \map \cos {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-2} - \cfrac 1 {a_{n-1}}} }}} }$

where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$ and
 * A terminal $a_{n-1} = \cos \theta$ term.