If $x$ = tanθ + cotθ and $y$ = tanθ − cotθ, then $x^2-y^2$ equals:

Factor rather than expand: $x^2-y^2=(x+y)(x-y)$ = (2tanθ)(2cotθ) = 4·tanθ·cotθ = 4, because tanθ and cotθ are reciprocals. So θ disappears completely. Check at θ = 37°: $x=0.7536+1.3270=2.0806$ and $y=-0.5734$, and 4.329 − 0.329 = 4.000. Expanding and mis-signing the cross term 2·tanθ·cotθ gives 0 or 2, and forgetting that the second bracket is 2cotθ rather than 2tanθ gives 4ta$n^2$θ.