 Solution

For a periodic function use the modulus of continuityOn the interval, take the supremum over pairs of points whose distance is at most $\delta$. The mean value theorem applied to cosine, whose derivative has absolute value at most one, givesBoth cosine values lie in $[-1,1]$, so for $|h|\le \delta$,Taking the supremum provesThe cosine substitution for polynomial approximation is also a linear isometry in the supremum norm, since cosine maps a full period onto $[-1,1]$; its image consists of even continuous periodic functions.