I'm always forgetting these even though
I need them from time to time. Fortunately, I have ben able to re-derive
them as needed.
We start both derivations with a given angle a. Let
b = a/2. Thus, 2b=a
Cosine
From that,
cos a = cos 2b
= cos2b - sin2b
= cos2b - sin2b
+ 1 - 1
= cos2b - sin2b
+ (cos2b + sin2b) - 1
= 2cos2b - 1
so
2cos2b = cos a + 1
cos2b = (cos a + 1) / 2
cos b = sqrt ((cos a + 1) / 2)
that is,
cos (a/2) = sqrt ((cos a + 1) / 2)
Sine
From above, we note that
cos2b = (cos a + 1) / 2
so
cos2b - 1 = (cos a + 1) / 2 - 1
cos2b - cos2b - sin2b = (cos
a + 1) / 2 - 2 / 2
-sin2b = (cos a + 1 - 2) / 2
= (cos a - 1) / 2
sin2b = (1 - cos a) / 2
sin b = sqrt ((1 - cos a) / 2)
that is,
sin (a/2) = sqrt ((1 - cos a) / 2)
The other formulas are easily derived from the sine and cosine formulas.
tan (a/2) = sin (a/2)/cos(a/2) = sqrt ((1 - cos a)/(1+cos a))
cot (a/2) = cos (a/2)/sin(a/2) = sqrt ((1+ cos a)/(1-cos a))
sec (a/2) = 1/cos(a/2) = sqrt (2/(1+cos a))
csc (a/2) = 1/sin(a/2) = sqrt (2/(1-cos a))