\section{Sine of x minus Cosine of x/Cosine Form}
Tags: Sine Function, Cosine Function

\begin{theorem}
:$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$
where $\sin$ denotes sine and $\cos$ denotes cosine.
\end{theorem}

\begin{proof}
{{begin-eqn}}
{{eqn | l = \sin x - \cos x
      | r = \sqrt 2 \, \map \sin {x - \dfrac \pi 4}
      | c = Sine of x minus Cosine of x: Sine Form
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x - \dfrac \pi 4} }
      | c = Cosine of Complement equals Sine
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - x + \dfrac \pi 4}
      | c = simplifying
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac {3 \pi} 4 - x}
      | c = simplifying
}}
{{eqn | r = \sqrt 2 \, \map \cos {x - \frac {3 \pi} 4}
      | c = Cosine Function is Even
}}
{{end-eqn}}
{{qed}}
Category:Sine Function
Category:Cosine Function
\end{proof}

