We should start from the given
#f(theta)=cos^2 (theta/4)+cot(theta/2)+sin^2(theta/2)#
Start with the first term using Half-Angle formula for cosine
#cos (A/2)=sqrt((1+cos A)/2)#
#color(red)(cos^2 (theta/4)=(1+cos (theta/2))/2=(1+sqrt((1+cos theta)/2))/2)#
For the second term using Half-Angle formulas for sine and cosine
#cot (A/2)=cos (A/2)/sin (A/2)=sqrt((1+cos A)/2)/sqrt((1-cos A)/2)#
#cot (A/2)=sqrt((1+cos A)/(1-cos A))=sqrt(((1+cos A)/(1-cos A))*((1+cos A))/((1+cos A)))#
#cot (A/2)=sqrt((1+cos A)^2/(1-cos^2 A))=sqrt((1+cos A)^2/(sin^2 A))#
#cot (A/2)=(1+cos A)/sin A#
therefore
#color(red)(cot (theta/2)=(1+cos theta)/sin theta)#
For the third term, using Half-Angle Formulas for Sine
#sin (A/2)=sqrt((1-cos A)/2)#
#sin^2 (A/2)=(1-cos A)/2#
therefore
#color(red)(sin^2 (theta/2)=(1-cos theta)/2)#
For the final answer
#f(theta)=cos^2 (theta/4)+cot(theta/2)+sin^2(theta/2)#
#color(red)(f(theta)=(1+sqrt((1+cos theta)/2))/2+(1+cos theta)/sin theta+(1-cos theta)/2)#
God bless ....I hope the explanation is useful.
