Theorem: Sine of Integer Multiple of Argument/Formulation 9

:$\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
:$r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
:$a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \theta & : \text {$k$ is odd and $k < n - 1$} \\ \sin \theta & : k = n - 1 \end {cases}$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
      | r = \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta}  + \map \sin {\paren {n - 2 } \theta} 
      | c = Sine of Integer Multiple of Argument/Formulation 6
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { \paren {2 \sin \theta }  + \frac {\map \sin {\paren {n - 2 } \theta} } {\map \cos {\paren {n - 1 } \theta} } }
      | c = Factor out $\map \cos {\paren {n - 1 } \theta}$
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {\cfrac {\map \cos {\paren {n - 1 } \theta} } {\map \sin {\paren {n - 2 } \theta} } } }
      | c = Move the numerator to the denominator
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {\cfrac {\paren {-2 \sin \theta } \map \sin {\paren {n - 2 } \theta}  + \map \cos {\paren {n - 3 } \theta}  } {\map \sin {\paren {n - 2 } \theta} } } }
      | c = Cosine of Integer Multiple of Argument/Formulation 6
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac {\map \cos {\paren {n - 3 } \theta} } {\map \sin {\paren {n - 2 } \theta} } } }
      | c = Simplify expression
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {\cfrac {\map \sin {\paren {n - 2 } \theta} } {\map \cos {\paren {n - 3 } \theta} } } } }
      | c = Move the numerator to the denominator
}}
{{eqn | ll = \leadsto
      | l = \frac {\map \sin {n \theta} } {\map \cos {\paren {n - 1 } \theta} }
      | r = \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {\cfrac {\map \sin {\paren {n - 2 } \theta} } {\map \cos {\paren {n - 3 } \theta} } } } }
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \frac {\map \sin {\paren {n - \paren {2 k} } \theta} } {\map \cos {\paren {n - \paren {2 k + 1 } } \theta} }
      | r = \paren {2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {\cfrac {\map \sin {\paren {n - 2 \paren {k + 1 } } \theta} } {\map \cos {\paren {n - \paren {2 \paren {k + 1} + 1} } \theta} } } } } 
      | c = $n \to \paren {n - 2 k}$
}}
{{end-eqn}}
Therefore, the terminal denominator will be:
{{begin-eqn}}
{{eqn | r = \paren {2 \sin \theta} + \frac {\map \sin {\paren {n - \paren {2k } } \theta} } {\map \cos {\paren {n - \paren {2k - 1 } } \theta} }
      | c = 
}}
{{end-eqn}}
Assume $n$ even:
$n:= 2k$
{{begin-eqn}}
{{eqn | r = \paren {2 \sin \theta} + \frac {\map \sin {\paren {2k - \paren {2k } } \theta} } {\map \cos {\paren {2k - \paren {2k - 1 } } \theta} }
      | c = 
}}
{{eqn | r = \paren {2 \sin \theta} + \frac {\map \sin {0} } {\map \cos {\theta} }
      | c = 
}}
{{eqn | r = \paren {2 \sin \theta}
      | c = Sine of Zero is Zero
}}
{{end-eqn}}
Assume $n$ odd:
$n:= 2k - 1$
{{begin-eqn}}
{{eqn | r = \paren {2 \sin \theta} + \frac {\map \sin {\paren {2k - 1 - \paren {2k } } \theta} } {\map \cos {\paren {2k - 1 - \paren {2k - 1 } } \theta} }
      | c = 
}}
{{eqn | r = \paren {2 \sin \theta} + \frac {\map \sin {-\theta} } {\map \cos 0}
      | c = 
}}
{{eqn | r = \paren {2 \sin \theta} - \frac {\map \sin {\theta} } {\map \cos 0}
      | c = Sine Function is Odd
}}
{{eqn | r = \paren {\sin \theta}
      | c = Cosine of Zero is One
}}
{{end-eqn}}
Therefore:
:$\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
:$r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
:$a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \theta & : \text {$k$ is odd and $k < n - 1$} \\ \sin \theta & : k = n - 1 \end {cases}$
{{qed}}