If $x$ = cosecθ + cotθ, then $(x^2-1)/(x^2+1)$ equals:

Because cose$c^2$θ − co$t^2$θ = 1, the conjugate is cosecθ − cotθ $=1/x$. Adding and subtracting gives cosecθ = $(x^2+1)/(2x)$ and cotθ = $(x^2-1)/(2x)$. Their quotient is cotθ/cosecθ = cosθ. Check at θ = 60°: $x=1.1547+0.5774=1.7321$, so (3 − 1)/(3 + 1) = 0.5 = cos60°.