I need help with Dudley problem 4.2.6

Theorem 4.2.8 in Dudley’s Real Analysis and Probability states:

Given a set $X$, a measurable space ($Y$, $mathcal{B}$), and a function
$T$ from $X$ into $Y$ , a real-valued function $f$ on $X$ is $T^{−1}[mathcal{B}]$ measurable on $X$ if and only if $f = g circ T$ for some $mathcal{B}$-measurable function $g$ on $Y$ .

And then here is problem 4.2.6:

In Theorem 4.2.8, let $X = mathbb{R}$, let $Y$ be the unit circle in $mathbb{R}^2$: $Y := {(x, y):
x^2 + y^2 = 1}$, and $mathcal{B}$ the Borel $sigma$-algebra on $Y$ . Let $T (u) := (cos u,sin u)$
for all $u in mathbb{R}$. Find which of the following functions $f$ on $mathbb{R}$ are $T^{−1}[mathcal{B}]$ measurable, and for those that are, find a function $g$ as in Theorem 4.2.8:

(a) $f (t) = cos(2t)$; (b) $f (t) = sin(t/2)$; (c) $f (t) = sin^2
(t/2)$.

So far, (a) and (c) are $T^{−1}[mathcal{B}]$measurable, as I can have $g = x^2 -y^2$ and $g = frac{1-x}{2}$ respectively, and as they are continuous, they are $mathcal{B}$-measurable (Am I right?).

I also think (b) is not $T^{−1}[mathcal{B}]$measurable, but I’m not sure how I can show that…

Any help would be greatly appreciated. Thank you.