To prove cotθ-½-sec2θ=1/(tan2θ+tanθ).
First, test the proposed identity: let θ=π/4; tanθ=cotθ=1; secθ=√2, sec2θ=2.
LHS=1-½-2=-3/2; RHS=1/(1+1)=½, so LHS≠RHS. The proposed identity is false.
Another interpretation of the question: To prove cotθ-½-sec2θ=(1/tan2θ)+tanθ.
RHS=1+1=2, so LHS≠RHS. The proposed identity is false.
Another interpretation of the question: To prove cotθ-1/(2-sec2θ)=1/(tan2θ+tanθ).
When θ=π/4, 2-sec2θ=0 so LHS is undefinable while RHS=½. Again, LHS≠RHS. The proposed identity is false.
Another interpretation of the question: To prove (cotθ-1)/2-sec2θ=1/(tan2θ+tanθ).
LHS=-2; RHS>0, LHS≠RHS. The proposed identity is false.
If there are no other interpretations of the question, the identity cannot be proved.