Start with the angle duplication formula $$ \tan (x) = {2 \tan (x/2) \over 1 - \tan^2 (x/2)}. $$ Cross-multiply and move terms around: $$ \tan (x) \tan^2 (x/2) + 2 \tan (x/2) = \tan (x) $$ Divide by $\tan (x)$ : $$ \tan^2 (x/2) + {2 \tan (x/2) \over \tan x} = 1 $$ Add $1 / \tan^2 (x)$ to both sides: $$ \tan^2 (x/2) + {2 \tan (x/2) \over \tan x} + {1 \over \tan^2 (x)} = 1 + {1 \over \tan^2 (x)} $$ Complete the square: $$ \left (\tan (x/2) + {1 \over \tan (x)} \right)^2 = 1 + {1 \over \tan^2 (x)} $$ Take a square root and move a term to obtain the half-angle formula: $$ \tan (x/2) = \sqrt{ 1 + {1 \over \tan^2 (x)} } - {1 \over \tan (x)} $$

To derive the other forms of the formula, we start by substituting $\sin (x) / \cos (x)$ for $\tan (x)$ : $$ \tan (x/2) = \sqrt{ 1 + {\cos^2 (x) \over \sin^2 (x)}} - {\cos (x) \over \sin (x)} $$ Put the stuff inside the square root over a common denominator: $$ \sqrt {\sin^2 (x) + \cos^2 (x) \over \sin^2 (x)} - {\cos (x) \over \sin (x)} $$ Recall that $\sin^2 (x) + \cos^2 (x) = 1$ . Hence, we may get rid of the square root: $$ {1 \over \sin x} - {\cos (x) \over \sin (x)} $$ Putting the terms over a common denominator, we obtain our formula: $$ \tan (x/2) = {1 - \cos (x) \over \sin (x)} $$

To obtain the next formula, multiply both numerator and denominator by $1 + \cos (x)$ : $$ {(1 - \cos (x)) (1 + \cos (x)) \over \sin (x) (1 + \cos (x))} $$ Multiply out the numerator and simplify: $$ {1 - \cos^2 (x) \over \sin (x) (1 + \cos (x))} $$ Note that the numerator equals $\sin^2 (x)$ : $$ {\sin^2 (x) \over \sin (x) (1 + \cos (x))} $$ Cancel a common factor of $\sin (x)$ to obtain the formula $$ \tan (x/2) = {\sin (x) \over 1 + \cos (x)} . $$

To obtain the last formula, multiply the previous two formulae: $$ \tan^2 (x/2) = {1 - \cos (x) \over \sin (x)}\cdot {\sin (x) \over 1 + \cos (x)} $$ Cancel the common factor of $\sin (x)$ : $$ \tan^2 (x/2) = {1 - \cos (x) \over 1 + \cos (x)} $$ Take the square root of both sides to obtain the formula $$ \tan{\frac{x}{2}}\; = \;\pm\sqrt{1 - \cos{x} \over 1 + \cos{x}}; $$ here the sign ($\pm$ ) has to be chosen according to the quadrant where the angle $\displaystyle\frac{x}{2}$ is.