# Double Angle Formulas

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

If we take alpha=t,beta=t then we will obtain following three formulas:

1. sin(2t)=2sin(t)cos(t),
2. cos(2t)=cos^2(t)-sin^2(t),
3. tan(2t)=(2tan(t))/(1-tan^2(t)).

Formula (3) is true when t!=pi/4+(pi k)/2, k in ZZ.

Above three formulas are called double angle formulas. With its help we can express sine, cosine, tangent of any argument in terms of trigonometric functions of half argument.

For example, following is true: sin(5t)=2sin((5t)/2)cos((5t)/2), cos(8t)=cos^2(4t)-sin^2(4t).

In many cases it is useful to use formulas "from right to left", i.e. we substitute expression 2sin(t)cos(t) with expression sin(2t), expression cos^2(t)-sin^2(t) with expression cos(2t), and expression (2tan(t))/(1-tan^2(t)) with expression tan(2t).

Example. Simplify following expression: tan(t)-cot(t).

tan(t)-cot(t)=(sin(t))/(cos(t))-(cos(t))/(sin(t))=(sin^2(t)-cos^2(t))/(sin(t)cos(t))=-(cos^2(t)-sin^2(t))/(1/2*2sin(t)cos(t))=-2(cos(2t))/(sin(2t))=-2cot(2t).