(1+cosx)/(1+secx)1+cosx1+secx
= (1+cosx)/(1+1/cosx)1+cosx1+1cosx
= (1+cosx)/((cosx+1)/cosx)1+cosxcosx+1cosx
= (1+cosx)xxcosx/(1+cosx)(1+cosx)×cosx1+cosx
= cosxcosx
Hence (1+cosx)/(1+secx)=cosx1+cosx1+secx=cosx for all values of xx
and for x=pi/3x=π3
(1+cosx)/(1+secx)=(1+cos(pi/3))/(1+sec(pi/3))1+cosx1+secx=1+cos(π3)1+sec(π3)
= (1+1/2)/(1+2)=(3/2)/3=3/2xx1/3=1/21+121+2=323=32×13=12
As cosx=cos(pi/3)=1/2cosx=cos(π3)=12
for x=pi/3x=π3, (1+cosx)/(1+secx)=cosx1+cosx1+secx=cosx