Half Angle Identities

Function	Formula	Description
sin(θ/2)	$\displaystyle\pm\sqrt{\frac{1 - \cos\theta}{2}}$	Plus or minus depends on quadrant of θ/2 - derived from cosine double angle
cos(θ/2)	$\displaystyle\pm\sqrt{\frac{1 + \cos\theta}{2}}$	Sign determined by which quadrant θ/2 falls in - always check the angle range
tan(θ/2)	$\displaystyle\pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$	Square root form - can also be expressed as (1-cosθ)/sinθ or sinθ/(1+cosθ)
tan(θ/2)	$\displaystyle\frac{1 - \cos\theta}{\sin\theta}$	Alternative form - no ambiguous sign, more practical for calculations
tan(θ/2)	$\displaystyle\frac{\sin\theta}{1 + \cos\theta}$	Second alternative form - equivalent to previous, choose based on given information
csc(θ/2)	$\displaystyle\pm\sqrt{\frac{2}{1 - \cos\theta}}$	Reciprocal of sine half-angle - undefined when cosθ = 1
sec(θ/2)	$\displaystyle\pm\sqrt{\frac{2}{1 + \cos\theta}}$	Reciprocal of cosine half-angle - undefined when cosθ = -1
cot(θ/2)	$\displaystyle\pm\sqrt{\frac{1 + \cos\theta}{1 - \cos\theta}}$	Square root form of cotangent - also equals (1+cosθ)/sinθ or sinθ/(1-cosθ)