In order to simplify what we have here, we have to use the Half-Angle identities and Sum and Difference formula.
First of all, let's simplify #cos(theta/4-pi/2)#. Knowing that
#cos(a-b)=cosacosb+sinasinb#
#=> cos(theta/4-pi/2)=cos(theta/4)cos(pi/2)+sin(theta/4)cos(pi/2)#
#cos(theta/4-pi/2) = cos(theta/4)*0 + sin(theta/4)*1=sin(theta/4)#
#:. cos(theta/4-pi/2) = sin(theta/4)#
The Half-Angle identities state that
#{(cos(x/2)=+-sqrt((1+cosx)/2)" [1]"),(sin(x/2)=+-sqrt((1-cosx)/2)" [2]"):}#
The sign of the function we wish to have is given by the quadrant the angle #x# is in.
Let us keep simplifying the second part of the function;
#"[2]" => sin(theta/4)=+-sqrt((1-cos(theta"/"2))/2)#
#"[1]" => +-sqrt((1-cos(theta"/"2))/2)=+-sqrt((1+-sqrt((1+cos(theta))/2))/2#
#color(blue)( :. sin(theta/4)=+-sqrt((1+-sqrt((1+cos(theta))/2))/2#
Now, let's pay our focus on #sec(theta"/"4)#.
Knowing that #secx=1/cosx#, we have
#sec(theta/4) = 1/cos(theta/4)#
#"[1]" => 1/cos(theta/4) = +-1/sqrt((1+cos(theta"/"2))/2)=+-sqrt(2/(1+cos(theta"/"2))#
#"[1]" => +-sqrt(2/(1+cos(theta"/"2))) = +-sqrt(2/(1+-sqrt((1+cos(theta))/2#
#color(blue)( :. sec(theta/4) = +-sqrt(2/(1+-sqrt((1+cos(theta))/2)#
Finally, we can write #f(theta)# in terms of the unit #theta# as
#color(blue)(f(theta) = +-sqrt(2/(1+-sqrt((1+cos(theta))/2))) +-sqrt((1+-sqrt((1+cos(theta))/2))/2#
Note: Because we had #1-cos(theta"/"2)# in the representation of #sin(theta"/" 4)# and #1+cos(theta"/"2)# in the representation of #sec(theta"/"4)#, whenever we have + in the first expression, we will have - in the second one, such as:
#sec(theta/4) = +-sqrt(2/(1color(red)+sqrt((1+cos(theta))/2)))#
We must have
#sin(theta/4) = +-sqrt((1color(red)-sqrt((1+cos(theta))/2))/2#
And vice versa.
