= Solution
Let $A=\langle X\rangle$ be the <quadratic variation>. We use bilinear <quadratic covariation> for the complex martingale: $\langle M,X\rangle=\langle\operatorname{Re}M,X\rangle+i\langle\operatorname{Im}M,X\rangle$. The <Itô formula> gives
$$
d(e^{-2i\theta X_t})=-2i\theta e^{-2i\theta X_t}\,dX_t-2\theta^2e^{-2i\theta X_t}\,dA_t.
$$
By the <Itô product rule>, the finite-variation part of $d(e^{-2i\theta X}M)$ is
$$
e^{-2i\theta X_t}\left(-2\theta^2 M_t\,dA_t-2i\theta\,d\langle M,X\rangle_t\right).
$$
Part (a) says the product is a martingale, so uniqueness of the continuous semimartingale decomposition makes this finite-variation part zero. For $\theta\ne0$, division gives
$$
\boxed{d\langle M,X\rangle_t=i\theta M_t\,d\langle X\rangle_t.}
$$
For $\theta=0$, $M\equiv1$ and both sides are zero, so the identity holds without exception.
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