Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative The problem:

Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is standard $\mathbb P$-Brownian motion.
Let $X = \{X_t\}_{t \in [0,T]}$ be a stochastic process where $X_t = W_t + \sin t$, and let $\mathbb Q$ be an equivalent probability measure s.t. $X$ is standard $\mathbb Q$-Brownian motion.
Give $\frac{d \mathbb Q}{d \mathbb P}$.

Girsanov Theorem:

Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is the standard $\mathbb P$-Brownian motion.
Let the Girsanov kernel $\{\theta_t\}_{t \in [0,T]}$ be a $\mathscr F_t$-adapted stochastic process s.t. $\int_0^T \theta_s^2 ds < \infty$ a.s. and $\{L_t\}_{t \in [0,T]}$ is a $( \mathscr F_t , \mathbb P)$ martingale where
$$L_t := \exp(-\int_0^t \theta_s dW_s - \frac 1 2 \int_0^t \theta_s^2 ds)$$
Let $\mathbb Q$ be the probability measure defined by
$$Q(A) = \int_A L_T dP \ \forall A \in \ \mathscr F$$
or $$L_T = \frac{d \mathbb Q}{d \mathbb P}$$
Then $\{W_t^Q\}_{t \in [0,T]}$ defined by
$$W_t^Q := W_t + \int_0^t \theta_s ds$$
is standard $\mathbb Q$-Brownian motion.


The solution given:
$$X_t = W_t + \int_0^t \cos s ds$$
Let $\theta_t = \cos t$:

*

*It is $\mathscr F_t$-adapted


*$\int_0^T \theta_s^2 ds < \infty$ a.s.


*$E[\exp(\frac 1 2 \int_0^T \theta_t^2 dt)] < \infty$
Then $\{L_t\}_{t \in [0,T]}$ is a $( \mathscr F_t , \mathbb P)$ martingale, by Novikov's condition, where
$$L_t := \exp(-\int_0^t \cos s dW_s - \frac 1 2 \int_0^t \cos^2 s ds)$$
Thus, by Girsanov's Theorem, we have
$$\frac{d\mathbb Q}{d\mathbb P} = L_T...?$$

How exactly does that last line follow?
What I find strange is that the Girsanov Theorem defines $\mathbb Q$ and then concludes $X_t$ is standard $\mathbb Q$-Brownian motion while the problem says there is some $\mathbb Q$ s.t. $X_t$ is standard $\mathbb Q$-Brownian motion and then asks about $\frac{d \mathbb Q}{d \mathbb P}$. Is the problem maybe stated wrong?
To say that $L_T$ is indeed the required density $\frac{d \mathbb Q}{d \mathbb P}$, I think we need to use the converse of the Girsanov Theorem (or here), or maybe the problem should instead give us $\frac{d \mathbb Q}{d \mathbb P}$ and then ask us to show that $L_T = \frac{d \mathbb Q}{d \mathbb P}$ possibly showing that $E[\frac{d \mathbb Q}{d \mathbb P} | \mathscr F_t] = L_t$ or some other route.

I tried something slightly different:
I define $\hat{\mathbb P}$ s.t.
$$L_T = \frac{d\hat{\mathbb P}}{d\mathbb P}$$
or
$$\hat{\mathbb P} = \int_A L_T d\mathbb P$$
It follows by Girsanov Theorem that $X_t$ is standard $\hat{\mathbb P}$-Brownian motion. Since we are given that there is some $\mathbb Q$ equivalent to $\mathbb P$ s.t. $X_t$ is also standard $\mathbb Q$-Brownian motion, it follows by the uniqueness of the Radon-Nikodym derivative that
$$\frac{d\hat{\mathbb P}}{d\mathbb P} = \frac{d\mathbb Q}{d\mathbb P}$$
$\therefore, \frac{d\mathbb Q}{d\mathbb P}$ is given by $L_T$.
Is that right? I think I'm missing a step somewhere.
So, is that indeed what the solution given is meant to be but just omitted pointing out uniqueness of the Radon-Nikodym derivative, if such justification is right?

Edit based on comment below and this: Even if Radon-Nikodym derivative is unique, $\mathbb Q$ may not be unique? If so, is it then that $\hat{\mathbb P}$ is merely a candidate for one of many possible $\mathbb Q$'s?
I think we conclude $\hat{\mathbb P} = \mathbb Q$ based on $X_t$ being standard Brownian motion under both measures. Is there a proposition for that? Uniqueness of Brownian motion measure or something?
 A: The R-N derivative process $Z_t:=d\Bbb Q|_{\mathcal F_t}/d\Bbb P|_{\mathcal F_t}$ is a strictly positive $\Bbb P$-martingale. As such it can be written as a stochatic exponential $\exp(M_t-{1\over 2}\langle M\rangle_t)$, with $M$ a $(\mathcal F_t^W,\Bbb P)$-local martingale. Thus $M$ admits a stochastic integral representation as $M_t=\int_0^t H_s\,dW_s$, with $H$ predicatable and $\int_0^T H_s^2\,ds<\infty$ $\Bbb P$-a.s. By Girsanov's theorem, the process $W_t-\int_0^t H_s\,ds$ is a $\Bbb Q$-local martingale. By hypothesis, $W_t+\sin t$ is also a $\Bbb Q$-local martingale. Subtracting we find that the process $\int_0^t H_sds+\sin t$ is a continuous $\Bbb Q$-local martingale that is also of finite variation. Consequently, $\int_0^t H_sds+\sin t=0$ for all $t\ge0$, $\Bbb Q$-a.s. (hence also $\Bbb P$-a.s.). It follows that $H_t(\omega)=-\cos t$ for $\Bbb P\otimes \lambda$-a.e $(\omega,t)\in \Omega\times[0,T]$. (Here $\lambda $ is Lebesgue measure.) In particular, $M_t=-\int_0^t\cos s\,dW_s$, and $d\Bbb Q/d\Bbb P=\exp(-\int_0^T\cos s\,dW_s-{1\over 2}\int_0^T\cos^2s\,ds)$.
