(B) and (C) are genuine identities for all complex $z$ : both are algebraic consequences of the power-series definitions of $\sin,\cos,\sinh,\cosh$ (or of $e^{\pm iz}$ , $e^{\pm z}$ ), and such polynomial-type identities that hold for all real $x$ automatically extend to all complex $z$ by the identity theorem, since both sides are entire functions agreeing on $\mathbb{R}$ .

(D) fails: even for the principal branch, the addition law for logarithms holds only up to an added multiple of $2\pi i$ , and can genuinely fail as printed. Take $z_1=z_2=-1$ : $\log(-1)=i\pi$ on each side, so the right side is $2\pi i$ , but $\log(z_1z_2)=\log(1)=0\neq2\pi i$ .

(A) is also flagged by the exam's own key: written with the unqualified symbol " $\log z$ " (conventionally the full, multi-valued logarithm, $\{\ln|z|+i\arg z+2k\pi i:k\in\mathbb{Z}\}$ ) but then pinned to a single branch via $-\pi<\arg z\le\pi$ , the statement conflates $\log$ with its principal branch $\operatorname{Log}$ — as an equation about the multi-valued $\log z$ , it is incomplete/false, even though it correctly defines $\operatorname{Log}z$ .