Double Angle Formula
Since quaternions use expressions like sin(t/2) and cos(t/2) it would be useful to have expressions for these in terms of sin(t) and cos(t)
As a starting point take the following trig functions:
sin(2A) = 2 sin(A) cos(A)
cos(2A) = 2 cos²(A)  1 = 1  2 sin²(A)
where:
 cos²(A) is shorthand notation for (cos(A))², that is, the square of cos
 sin²(A) is shorthand notation for (sin(A))², that is, the square of sine
Half Angle Formula

Graphical RepresentationWe can show these relationships graphically tan(θ)=opposite/adjacent = sin(θ)/(cos(θ)+1)

Derivation
In the above double angle formula we substitute t=2A to give:
1  2 sin²(t/2) = cos(t)
sin²(t/2) =0.5 (1 cos(t))
sin(t/2) =√(0.5 (1 cos(t)))
Similarly for cosine:
2 cos²(t/2)  1 = cos(t)
2 cos²(t/2) = 1 + cos(t)
cos(t/2) =√(0.5 (1+ cos(t)))