# Formulas for Adding and Subtracting Arguments

For any real numbers alpha and beta following formulas are true:

1. cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta),
2. cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta),
3. sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta),
4. sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta),
5. tan(alpha+beta)=(tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta)),
6. tan(alpha-beta)=(tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta)).

These formulas are called formulas for adding and subtracting arguments.

Fomula (5) holds when alpha,beta,alpha+beta are not equal pi/2+pik,k in ZZ (in other words when tangent is defined). Formula (6) holds when alpha,beta,alpha-beta are not equal pi/2+pik, k in ZZ.

Example 1 . Find sin(75^0).

We have that sin(75^0)=sin(30^0+45^0).

Using formula (3) with alpha=30^0,beta=45^0 we obtain that sin(30^0+45^0)=sin(30^0)cos(45^0)+cos(30^0)sin(45^0).

It is known that sin(30^0)=1/2,cos(30^0)=(sqrt(3))/2,cos(45^0)=sin(45^0)=(sqrt(2))/2.

Therefore, sin(75^0)=sin(30^0+45^0)=1/2*(sqrt(2))/2+(sqrt(3))/2*(sqrt(2))/2=(sqrt(2)+sqrt(6))/4.

Example 2. Find tan(pi/4+alpha) if tan(alpha)=3/4.

Using formula (5) and fact that tan(pi/4)=1 we have that

tan(pi/4+alpha)=(tan(pi/4)+tan(alpha))/(1-tan(pi/4)tan(alpha))=(1+tan(alpha))/(1-tan(alpha))=(1+3/4)/(1-3/4)=7.