For an angle θ with tanθ − cotθ = 3, the value of ta$n^2$θ + co$t^2$θ is:

$(tan\theta -cot\theta {)}^2=ta$$n^2$θ + $co$$t^2$θ − 2·tanθ·cotθ = $ta$$n^2$θ + $co$$t^2$θ − 2, so $ta$$n^2$θ + $co$$t^2$θ = 9 + 2 = 11. Because the cross term was subtracted, it has to be added back — running the sum-version reflex and taking 9 − 2 = 7 is the trap. Ignoring the cross term altogether leaves 9, and doubling the correction gives 13.