Question

Rewrite cot^2⁡〖x/4〗 in terms of cosine without power greater than one. Assume that x is in QI.

Answer 1

your half-angle formulas state that

tan(a/2) = (1-cosa)/sina = sina/(1+cosa)
That means that
cot^2(x/4) = sin(x/2)/(1-cos(x/2)) * (1 + cos(x/2))/sin(x/2)
= (1 + cos(x/2))/(1 - cos(x/2))
= (1 + cos(x/2))^2 / (1 - cos^2(x/2))
= (1 + 2cos(x/2) + (1 + cosx)/2) / (1 - cosx)/2
= (2 + 4cos(x/2) + 1 + cosx) / (1 - cosx)
= (3 + 4cos(x/2) + cosx) / (1 - cosx)
Maybe you can massage that some, but I don't see any way to get rid of that pesky cos(x/2)