We know that the formulas of Weierstrass or named also parametric formulas (obviously true in their domain of definition) are:
$$\tan \left(\frac \alpha2\right):=t, \quad \sin(\alpha)=\frac{2t}{1+t^2},\, \cos(\alpha)=\frac{1-t^2}{1+t^2} \tag 1$$
We suppose that we have these examples of identities:
$$\tan \left(\frac \alpha2\right)=\csc(\alpha)-\cot(\alpha) \tag 2$$ $$\tan \left(\frac \pi4-\frac \alpha2\right)=\frac{\cos(\alpha)}{1+\sin(\alpha)}\tag 3$$
I have suggest to my students to use the Weiestrass' formulas to have an equation in function of $t$ because there are not squares in $(2)$ and $(3)$ (or we not seen any square of a trigonometric functions in function of $\alpha/2$ - for example).
In fact I have given a suggestion to never use the formulas $(5)$,
$$\sin (\alpha)=\pm \sqrt{\frac{1-\cos(\alpha)}{2}}, \quad \cos (\alpha)=\pm \sqrt{\frac{1+\cos(\alpha)}{2}} \tag 5$$
because we not know the value of $\alpha$.
My question:
Do you think (for $(2)$ and $(3)$) is the best alternative to use Weierstrass formulas when we see goniometric operators as a function of half angle $\alpha/2$ and to $\alpha$ in the identities?