Let's talk about the answer a bit first. There are two angles on the unit circle whose cotangent is #13#. One is around #4.4^circ#, and another is that plus #180^circ#, call it #184.4^circ#. Each of those have two half angles, again separated by #180^circ.# The first one has half angles #2.2^circ# and #182.2^circ#, the second has half angles #92.2^circ# and #272.2^circ#, So there are really four half angles in question, with differing but related values for their trig functions.
We'll use the above angles as approximations so we have names for them.
Angles with cotangent of 13:
#text{Arc}text{cot} 13 approx 4.4^circ #
#180 ^circ + text{Arc}text{cot} 13 approx 184.4^circ #
Half angles:
#1/2 text{Arc}text{cot} 13 approx 2.2^circ #
#1/2 (360^circ + text{Arc}text{cot} 13) approx 182.2^circ #
#1/2 ( 180 ^circ + text{Arc}text{cot} 13) approx 92.2^circ #
#1/2 ( 360^circ + 180 ^circ + text{Arc}text{cot} 13) approx 272.2^circ #
OK, the double angle formulas for cosine are:
#cos(2a) = 2 cos^2 a - 1 = 1 - sin ^2 a #
so the relevant half angle formulas are
#sin a = pm sqrt{1/2(1-cos(2a))}#
#cos a = pm sqrt{1/2(1+cos(2a))}#
That's all preliminary. Let's do the problem.
We'll do the tiny angle first, #2.2^circ.# We see the rest of them are just multiples of #90^circ# above that, so we can get their trig functions from this first angle.
A cotangent of 13 is a slope of #1/13# so corresponds to a right triangle with opposite #1#, adjacent #13# and hypotenuse #sqrt{13^2 + 1^2}=sqrt{170}.#
#cos( text{Arc}text{cot} 13 ) = cos 4.4^circ = {13}/sqrt{170}#
#sin( text{Arc}text{cot} 13 ) = sin 4.4^circ = {1}/sqrt{170}#
Now we apply the half angle formulas. For our teeny angle in the first quadrant, we choose the positive signs.
#cos( 1/ 2text{Arc}text{cot} 13 ) = cos 2.2 ^circ = \sqrt{1/2(1 + cos(4.4^circ))} = sqrt{1/2 ( 1 + {13}/sqrt{170})}#
We could try to simplify and move the fractions outside the radical, but I'm just going to leave it here.
#sin( 1/ 2text{Arc}text{cot} 13 ) = sin 2.2 ^circ = \sqrt{1/2(1 - cos(4.4^circ))} = sqrt{1/2 ( 1 - {13}/sqrt{170})}#
The tangent half angle is the quotient of those, but it's easier to use
# tan(theta/2) = { sin theta}/{1 +cos theta} #
#tan( 1/ 2text{Arc}text{cot} 13 ) = tan 2.2^circ = {1/sqrt{170}}/ { 1 + {13}/sqrt{170}} = sqrt(170) - 13 #
OK, that's all the hard part, but let's not forget the other angles.
# cos 182.2 ^circ = - cos 2.2 ^circ = - sqrt{1/2 ( 1 + {13}/sqrt{170})}#
#sin 182.2^circ = -sin 2.2^circ = - sqrt{1/2 ( 1 - {13}/sqrt{170})}#
# tan 182.2^circ = tan 2.2^circ = sqrt(170) - 13 #
Now we have the remaining angles, which swap sine and cosine, flipping signs. We won't repeat the forms except for tangent.
# cos 92.2^circ = - sin 2.2 ^circ #
#sin 92.2^circ = cos 2.2^circ #
# tan 92.2^circ = -1/{tan 2.2^circ }= -13 - sqrt(170) #
# cos 272.2^circ = sin 2.2 ^circ #
#sin 272.2^circ = - cos 2.2^circ #
# tan 272.2^circ = tan 92.2^circ = -13 - sqrt(170) #
Phew.
