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))

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