For an acute angle θ, the expression (1 + sinθ)/(1 − sinθ) − (1 − sinθ)/(1 + sinθ) simplifies to:

The common denominator is (1 − sinθ)(1 + sinθ) = co$s^2$θ, and the numerator is $(1+sin\theta {)}^2-(1-sin\theta {)}^2=4$sinθ. So the expression is 4sinθ/co$s^2$θ = 4·(1/cosθ)·(sinθ/cosθ) = 4·secθ·tanθ. Check at θ = 30°: 3 − 1/3 = 2.667 and $4\times 1.1547\times 0.5774=2.667$. Cancelling only one factor of cosθ leaves 4tanθ, halving the numerator gives 2secθ·tanθ, and dropping the sinθ gives 4secθ.