Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Let $z_1=e^{iA}$, $z_2=e^{iB}$, $z_3=e^{iC}$. The given conditions imply $z_1+z_2+z_3=(\cos A+\cos B+\cos C)+i(\sin A+\sin B+\sin C)=0$.

Since $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$, it implies that $z_1,z_2,z_3$ form an equilateral triangle inscribed in the unit circle. Thus, the angles $A,B,C$ must be of the form $\theta ,\theta +2\pi /3,\theta +4\pi /3$ (in some order). For option (A): $z_1+z_2+z_3=0⟹(z_1+z_2+z_3)(\overline{z_1}+\overline{z_2}+\overline{z_3})=0$. This expands to $3+z_1\overline{z_2}+z_2\overline{z_1}+z_2\overline{z_3}+z_3\overline{z_2}+z_3\overline{z_1}+z_1\overline{z_3}=0$. So $3+2\cos (A-B)+2\cos (B-C)+2\cos (C-A)=0$. Since the angles differ by $2\pi /3$ or $4\pi /3$, $\cos (A-B)=\cos (B-C)=\cos (C-A)=-1/2$. Thus, $3+2(-1/2-1/2-1/2)=3-3=0$, which is consistent. So $\cos (A-B)+\cos (B-C)+\cos (C-A)=-3/2$. (A) is correct.

For option (B): If $z_1+z_2+z_3=0$ and $|z_1|=|z_2|=|z_3|=1$, then $z_1^2+z_2^2+z_3^2=0$. This is because $z_1{z}_2+z_2{z}_3+z_3{z}_1=z_1{z}_2{z}_3(1/z_1+1/z_2+1/z_3)=z_1{z}_2{z}_3(\overline{z_1}+\overline{z_2}+\overline{z_3})=z_1{z}_2{z}_3(\overline{z_1+z_2+z_3})=0$. Since $(z_1+z_2+z_3{)}^2=z_1^2+z_2^2+z_3^2+2(z_1{z}_2+z_2{z}_3+z_3{z}_1)=0$, we have $z_1^2+z_2^2+z_3^2=0$. This means $e^{i2A}+e^{i2B}+e^{i2C}=0$, so $(\cos 2A+\cos 2B+\cos 2C)+i(\sin 2A+\sin 2B+\sin 2C)=0$. Hence $\cos 2A+\cos 2B+\cos 2C=0$. (B) is correct. For option (C): If $z_1+z_2+z_3=0$, then $z_1^3+z_2^3+z_3^3=3z_1{z}_2{z}_3$. This is a standard algebraic identity. So $e^{i3A}+e^{i3B}+e^{i3C}=3e^{i(A+B+C)}$. Taking the imaginary parts, $\sin 3A+\sin 3B+\sin 3C=3\sin (A+B+C)$. (C) is correct.

For option (D): The identity $\tan A+\tan B+\tan C=\tan A\tan B\tan C$ holds if $A+B+C=n\pi$. However, $A,B,C$ are of the form $\theta ,\theta +2\pi /3,\theta +4\pi /3$, so $A+B+C=3\theta +2\pi$. This sum is not necessarily an integer multiple of $\pi$. For example, if $\theta =\pi /6$, then $A+B+C=5\pi /2$, for which $\tan (A+B+C)$ is undefined, and $\tan C=\tan (3\pi /2)$ is also undefined. Thus, (D) is not generally correct.