We will use the following Properties:
color(blue)(sin(x+y)=sinxcosy+cosxsinysin(x+y)=sinxcosy+cosxsiny
color(blue)(cos(x+y)=cosxcosy-sinxsinycos(x+y)=cosxcosy−sinxsiny
color(blue)(sin2x=2sinxcosxsin2x=2sinxcosx
color(blue)(cos2x=cos^2x-sin^2xcos2x=cos2x−sin2x
color(blue)(sin^2x+cos^2x=1sin2x+cos2x=1
LHS: tan(x/2 + pi/4)LHS:tan(x2+π4)
=sin(x/2+pi/4)/(cos(x/2+pi/4)=sin(x2+π4)cos(x2+π4)
=(sin (x/2)cos(pi/4)+cos(x/2)sin(pi/4))/(cos(x/2)cos(pi/4)-sin(x/2)sin(pi/4))=sin(x2)cos(π4)+cos(x2)sin(π4)cos(x2)cos(π4)−sin(x2)sin(π4)
=(1/sqrt2 sin(x/2)+1/sqrt2 cos (x/2))/(1/sqrt2cos(x/2)-1/sqrt2sin(x/2)=1√2sin(x2)+1√2cos(x2)1√2cos(x2)−1√2sin(x2)
=(1/sqrt2( sin(x/2)+ cos (x/2)))/(1/sqrt2(cos(x/2)-sin(x/2))=1√2(sin(x2)+cos(x2))1√2(cos(x2)−sin(x2))->factor out common factor
=(cancel(1/sqrt2)( sin(x/2)+ cos (x/2)))/(cancel(1/sqrt2)(cos(x/2)-sin(x/2))-> cancel common factor
=( sin(x/2)+ cos (x/2))/(cos(x/2)-sin(x/2))
=( sin(x/2)+ cos (x/2))/(cos(x/2)-sin(x/2)) * (cos(x/2)+sin(x/2))/(cos(x/2)+sin(x/2))->multiply by conjugate
=(sin^2(x/2)+2sin(x/2)cos(x/2)+cos^2(x/2))/(cos^2(x/2)-sin^2(x/2))
=([sin^2(x/2)+cos^2(x/2)]+2sin(x/2)cos(x/2))/(cos^2(x/2)-sin^2(x/2))
=(1+2sin(x/2)cos(x/2))/(cos^2(x/2)-sin^2(x/2))
=(1+sin 2(x/2))/(cos 2(x/2))
=(1+sin cancel2(x/cancel2))/(cos cancel2(x/cancel2))
=(1+sin x)/cos x
=1/cosx+sin x/cos x
=secx + tanx
=RHS
