.
#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#
