Double Angle Formulas

We already know following 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)`.