angle multiplication and division formulae for tangent

From the angle addition formula for the tangent, we may derive formulaefor tangents of multiples of angles:

	$\\displaystyle\\tan(2x)$	$\\displaystyle={2\\tan x\\over 1-\\tan^{2}x}$
	$\\displaystyle\\tan(3x)$	$\\displaystyle={3\\tan x-\\tan^{3}x\\over 1-3\\tan^{2}x}$
	$\\displaystyle\\tan(4x)$	$\\displaystyle={4\\tan x-3\\tan^{3}x\\over 1-6\\tan^{2}x+\\tan^{4}x}$

These formulae may be derived from a recursion. Write $\\tan x=w$ andwrite $\\tan(nx)=u_{n}/v_{n}$ where the $u$ ’s and the $v$ ’s arepolynomials in $w$ . Then we have the initial values $u_{1}=w$ and $v_{1}=1$ and the recursions

	$\\displaystyle u_{n+1}$	$\\displaystyle=u_{n}+wv_{n}$
	$\\displaystyle v_{n+1}$	$\\displaystyle=v_{n}-wu_{n},$

which follow from the addition formula. Moreover, if we know the tangentof an angle and are interested in finding the tangent of a multiple ofthat angle, we may use our recursions directly without first having to derivethe multiple angle formulae. From these recursions, one may show that the $u$ ’s will only involve odd powers of $w$ and the $v$ ’s will only involveeven powers of $w$ .

Proceeding in the opposite direction, one may consider bisecting an angle.Solving for $\\tan x$ in the duplication formula above, one arrives at thefollowing half-angle formula:

\\tan\\left({x\\over 2}\\right)=\\sqrt{1+{1\\over\\tan^{2}x}}-{1\\over\\tan x}

Expressing the tangent in terms of sines and cosines and simplifying, onefinds the following equivalent formulae:

\\tan\\left({x\\over 2}\\right)={1-\\cos x\\over\\sin x}={\\sin x\\over 1+\\cos x}=\\pm%\\sqrt{1-\\cos x\\over 1+\\cos x}