We know cos(alpha+beta)=cosalphacosbeta-sinalphasinbetacos(α+β)=cosαcosβ−sinαsinβ
and cos(alpha-beta)=cosalphacosbeta+sinalphasinbetacos(α−β)=cosαcosβ+sinαsinβ
Adding two we get cos(alpha+beta)+cos(alpha-beta)=2cosalphacosbetacos(α+β)+cos(α−β)=2cosαcosβ or
cosalphacosbeta=1/2{cos(alpha+beta)+cos(alpha-beta)}cosαcosβ=12{cos(α+β)+cos(α−β)}
Hence cos((3pi)/2)cos((13pi)/8)=1/2{cos((3pi)/2+(13pi)/8)+cos((3pi)/2-(13pi)/8)}cos(3π2)cos(13π8)=12{cos(3π2+13π8)+cos(3π2−13π8)}
= 1/2{cos((12pi)/8+(13pi)/8)+cos((12pi)/8-(13pi)/8)}12{cos(12π8+13π8)+cos(12π8−13π8)}
= 1/2{cos((25pi)/8)+cos(-pi/8)}12{cos(25π8)+cos(−π8)}
= 1/2{cos(2pi+pi+pi/8)+cos(pi/8)}12{cos(2π+π+π8)+cos(π8)}
= 1/2{-cos(pi/8)+cos(pi/8)}=012{−cos(π8)+cos(π8)}=0