p = (((Cos[t] + Sin[t]))^10)

q = Expand[p]

Factor[q]

Variables[p]

Collect[(Expand[
              p] //. {Cos[t]^k_ /; k > 1 -> 
                Expand[Cos[t]^(k - 2)*(1 - Sin[t]^2)]}) // Expand, 
      Cos[t]]

PolynomialRemainder[(((x + y))^10), (x^2) + (y^2) - 1, x]

TrigReduce[p]