# Trigonometrics function of Sum and difference of angles

## Trigonometrics function of Sum and difference of angles

### Sin and cos function

1. $cos(A+B)=cos(A)cos(B)-sin(A)sin(B)$
2. $cos(A-B)=cos(A)cos(B)+sin(A)sin(B)$
3. $cos(\frac {\pi}{2} -A)=sin(A)$
4. $sin(\frac {\pi}{2} -A)=cos(A)$
5. $sin(A+B)=sin(A)cos(B)+sin(B)cos(A)$
6. $sin(A-B)=sin(A)cos(B)-sin(B)cos(A)$
Similary we can have defined other sin and cos sum and differences

### Tan and cot functions

1. If none of the angles x, y and (x + y) is an odd multiple of π/2
$tan(A+B)=\frac{tan(A)+tan(B)}{1-tan(A)tan(B)}$
$tan(A-B)=\frac{tan(A)-tan(B)}{1+tan(A)tan(B)}$
2. If none of the angles x, y and (x + y) is an multiple of π
$cot(A+B)=\frac{cot(A)cot(B)-1}{cot(A)+cot(B)}$
$cot(A-B)=\frac{cot(A)cot(B)+1}{cot(B)-cot(A)}$
Now lets explore the multiple of x. These all can be proved from above equations
Double of x
$cos2x=cos^{^{2}}x-sin^{^{2}}x=2cos^{^{2}}x-1=1-2sin^{^{2}}x=\frac{1-tan^{^{2}}x}{1+tan^{^{2}}x}$
$sin2x=2cos(x)sin(x)=\frac{2tan(x)}{1+tan^{^{2}}x}$
$tan2x=\frac{2tan(x)}{1-tan^{^{2}}x}$
Triple of x
$sin3x=3sin(x)-4sin^{3}x$
$cos3x=4cos^{3}x-3cos(x)$

$tan(3x)=\frac{3tanx-tan^{^{3}}x}{1-3tan^{^{2}}x}$
Some other Important functions
• $cos(A)+cos(B)=2cos\frac{A+B}{2}cos\frac{A-B}{2}$
• $cos(A)-cos(B)=-2sin\frac{A+B}{2}sin\frac{A-B}{2}$
• $sin(A)+sin(B)=2sin\frac{A+B}{2}cos\frac{A-B}{2}$
• $sin(A)-sin(B)=2cos\frac{A+B}{2}sin\frac{A-B}{2}$