Ucale

# Sum of sines / cosine in terms of products

Values of  sin(A+B) and cos(A+B) in terms of their product is given by

1. $\displaystyle \sin { A } +\sin { B } =2\sin { \left( \frac { A+B }{ 2 } \right) } \cos { \left( \frac { A-B }{ 2 } \right) }$
2. $\displaystyle \sin { A } -\sin { B } =2\cos { \left( \frac { A+B }{ 2 } \right) } \sin { \left( \frac { A-B }{ 2 } \right) }$
3. $\displaystyle \cos { A } +\cos { B } =2\cos { \left( \frac { A-B }{ 2 } \right) } \cos { \left( \frac { A-B }{ 2 } \right) }$
4. $\displaystyle \cos { A } -\cos { B } =2\cos { \left( \frac { A+B }{ 2 } \right) } \cos { \left( \frac { B-A }{ 2 } \right) }$
5. $\displaystyle \tan { A } +\tan { B } =\frac { \sin { \left( A+B \right) } }{ \cos { A } \cos { B } } \quad where\quad A,B\neq n\pi +\frac { \pi }{ 2 }$
6. $\displaystyle \tan { A } -\tan { B } =\frac { \sin { \left( A-B \right) } }{ \cos { A } \cos { B } } \quad where\quad A,B\neq n\pi +\frac { \pi }{ 2 }$
7. $\displaystyle \cot { A } +\cot { B } =\frac { \sin { \left( A+B \right) } }{ \sin { A } \sin { B } } \quad where\quad A,B\neq n\pi \quad n=z$
8. $\displaystyle \cot { A } -\cot { B } =\frac { \sin { \left( B-A \right) } }{ \sin { A } \sin { B } } \quad where\quad A,B\neq n\pi \quad n=z$

Conversely

1. 2 sin A cos B = sin (A+ B) + sin (A – B)
2. 2 cos A sin B = sin (A + B) – sin (A – B)
3. 2 cos A cos B = cos (A + B) + cos (A – B)
4. 2 sin A sin B = cos (A – B) – cos (A + B)
February 22, 2019
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