.
sec(270^@-theta)sec(90^@-theta)-tan(270^@-theta)tan(90^@+theta)=-1sec(270∘−θ)sec(90∘−θ)−tan(270∘−θ)tan(90∘+θ)=−1
1/cos(270^@-theta)*1/cos(90^@-theta)-sin(270^@-theta)/cos(270^@-theta)*sin(90^@+theta)/cos(90^@+theta)=-11cos(270∘−θ)⋅1cos(90∘−θ)−sin(270∘−θ)cos(270∘−θ)⋅sin(90∘+θ)cos(90∘+θ)=−1
We have the following four identities:
sin(alpha+beta)=sinalphacosbeta+cosalphasinbetasin(α+β)=sinαcosβ+cosαsinβ
sin(alpha-beta)=sinalphacosbeta-cosalphasinbetasin(α−β)=sinαcosβ−cosαsinβ
cos(alpha+beta)=cosalphacosbeta-sinalphasinbetacos(α+β)=cosαcosβ−sinαsinβ
cos(alpha-beta)=cosalphacosbeta+sinalphasinbetacos(α−β)=cosαcosβ+sinαsinβ
Therefore,
cos(270^@-theta)=cos270^@costheta+sin270^@sintheta=(0)costheta+(-1)sintheta=0-sintheta=-sinthetacos(270∘−θ)=cos270∘cosθ+sin270∘sinθ=(0)cosθ+(−1)sinθ=0−sinθ=−sinθ
cos(90^@-theta)=cos90^@costheta+sin90^@sintheta=(0)costheta+(1)sintheta=0+sintheta=sinthetacos(90∘−θ)=cos90∘cosθ+sin90∘sinθ=(0)cosθ+(1)sinθ=0+sinθ=sinθ
sin(270^@-theta)=sin270^@costheta-cos270^@sintheta=(-1)costheta-(0)sintheta=-costheta-0=-costhetasin(270∘−θ)=sin270∘cosθ−cos270∘sinθ=(−1)cosθ−(0)sinθ=−cosθ−0=−cosθ
sin(90^@+theta)=sin90^@costheta+cos90^@sintheta=(1)costheta+(0)sintheta=costheta+0=costhetasin(90∘+θ)=sin90∘cosθ+cos90∘sinθ=(1)cosθ+(0)sinθ=cosθ+0=cosθ
cos(90^@+theta)=cos90^@costheta-sin90^@sintheta=(0)costheta-(1)sintheta=0-sintheta=-sinthetacos(90∘+θ)=cos90∘cosθ−sin90∘sinθ=(0)cosθ−(1)sinθ=0−sinθ=−sinθ
Now, let's substitute all the pieces:
1/(-sintheta)*1/sintheta-(-costheta)/(-sintheta)*costheta/(-sintheta)=-1/sin^2theta+cos^2theta/sin^2theta=1−sinθ⋅1sinθ−−cosθ−sinθ⋅cosθ−sinθ=−1sin2θ+cos2θsin2θ=
(-1+cos^2theta)/sin^2theta=(-(1-cos^2theta))/sin^2theta=(-sin^2theta)/sin^2theta=-1−1+cos2θsin2θ=−(1−cos2θ)sin2θ=−sin2θsin2θ=−1
