Werner Formulas/Sine by Sine/Corollary 1

Corollary to Werner Formula for Sine by Sine

$\map \sin {A + B} \map \sin {A - B} = \paren {\sin A + \sin B} \paren {\sin A - \sin B}$

Proof

\(\ds \map \sin {A + B} \map \sin {A - B}\)

\(=\)

\(\ds \dfrac {\map \cos {\paren {A + B} - \paren {A - B} } - \map \cos {\paren {A + B} + \paren {A - B} } } 2\)

Werner Formula for Sine by Sine, with $\alpha = A + B$ and $\beta = A - B$

\(\ds \)

\(=\)

\(\ds \dfrac {\cos 2 B - \cos 2 A} 2\)

simplification

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {\paren {1 - 2 \sin^2 B} - \paren {1 - 2 \sin^2 A} } } 2\)

Double Angle Formula for Cosine: Corollary $2$

\(\ds \)

\(=\)

\(\ds \sin^2 A - \sin^2 B\)

simplification

\(\ds \)

\(=\)

\(\ds \paren {\sin A + \sin B} \paren {\sin A - \sin B}\)

Difference of Two Squares

$\blacksquare$