.
sec(270^@-theta)sec(90^@-theta)-tan(270^@-theta)tan(90^@+theta)=-1
1/cos(270^@-theta)*1/cos(90^@-theta)-sin(270^@-theta)/cos(270^@-theta)*sin(90^@+theta)/cos(90^@+theta)=-1
We have the following four identities:
sin(alpha+beta)=sinalphacosbeta+cosalphasinbeta
sin(alpha-beta)=sinalphacosbeta-cosalphasinbeta
cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta
cos(alpha-beta)=cosalphacosbeta+sinalphasinbeta
Therefore,
cos(270^@-theta)=cos270^@costheta+sin270^@sintheta=(0)costheta+(-1)sintheta=0-sintheta=-sintheta
cos(90^@-theta)=cos90^@costheta+sin90^@sintheta=(0)costheta+(1)sintheta=0+sintheta=sintheta
sin(270^@-theta)=sin270^@costheta-cos270^@sintheta=(-1)costheta-(0)sintheta=-costheta-0=-costheta
sin(90^@+theta)=sin90^@costheta+cos90^@sintheta=(1)costheta+(0)sintheta=costheta+0=costheta
cos(90^@+theta)=cos90^@costheta-sin90^@sintheta=(0)costheta-(1)sintheta=0-sintheta=-sintheta
Now, let's substitute all the pieces:
1/(-sintheta)*1/sintheta-(-costheta)/(-sintheta)*costheta/(-sintheta)=-1/sin^2theta+cos^2theta/sin^2theta=
(-1+cos^2theta)/sin^2theta=(-(1-cos^2theta))/sin^2theta=(-sin^2theta)/sin^2theta=-1