trignometry prove cotA-1/2- sec^2A=1/tan^2A+tanA

Question

prove cot theta-1/2-sec^2 theta=1/tan^2theta+tan theta

Answer 1

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.