These formulas are as follows:
sinθ=1+tan22θ2tan2θ,cosθ=1+tan22θ1−tan22θ,tanθ=1−tan22θ2tan2θ.
We will use the double-angle formula to prove this:
sinθtanθcosθ=2sin2θcos2θ=cos22θ+sin22θ2sin2θcos2θ=1+tan22θ2tan2θ,=1−tan22θ2tan2θ,=tanθsinθ=1−tan22θ2tan2θ1+tan22θ2tan2θ=1+tan22θ1−tan22θ. □