If tanθ = 5/12 and θ is acute, then (1 + sinθ)(1 − sinθ) ÷ [(1 + cosθ)(1 − cosθ)] equals:

Each bracket pair is a difference of squares: (1 + sinθ)(1 − sinθ) = co$s^2$θ and (1 + cosθ)(1 − cosθ) = si$n^2$θ. The quotient is therefore co$t^2$θ = $(12/5{)}^2=144/25$. Taking the ratio the other way up gives ta$n^2$θ = 25/144, forgetting to square gives 12/5, and putting the hypotenuse over the opposite side gives cose$c^2$θ = 169/25.