Let $\left\{X_t\right\}$ be the process defined by $$ X_t=\sum_{j=1}^nA(\lambda_j)e^{it\lambda_j} $$ in which $-\pi<\lambda_1<\lambda_2<\ldots <\lambda_n=\pi$ and $A(\lambda_1),\ldots,A(\lambda_n)$ are uncorrelated complex-valued random coefficients (possibly zero) such that $$ E(A(\lambda_j))=0,~j=1,\ldots,n,\\ E(A(\lambda_j)\overline{A(\lambda_j)})=\sigma_j^2,~j=1,\ldots,n. $$
Task (1.) Show that $\left\{X_t\right\}$ is real-valued if and only if $\lambda_j=-\lambda_{n-j}$ and $A(\lambda_j)=\overline{A(\lambda_{n-j})}, j=1,\ldots,n-1$ and $A(\lambda_n)$ is real-valued.
(2.) Show that $\left\{X_t\right\}$ then satisfies $$ X_t=\sum_{j=1}^n (C(\lambda_j)\cos (t\lambda_j)-D(\lambda_j)\sin (t\lambda_j)), $$ where $A(\lambda_j)=C(\lambda_j)+iD(\lambda_j),~j=1,\ldots,n$ and $D(\lambda_n)=0$.
Using that $$ A(\lambda_j)=C(\lambda_j)+i\cdot D(\lambda_j) $$ and the identity $$ e^{it\lambda_j}=\cos(t\lambda_j)+i\cdot\sin(t\lambda_j), $$ I calculated that $$ X_t=\sum_{j=1}^n (C(\lambda_j)\cos(t\lambda_j)-D(\lambda_j)\sin(t\lambda_j))+i\cdot\sum_{j=1}^n (C(\lambda_j)\sin(t\lambda_j)+D(\lambda_j)\cos(t\lambda_j)) $$
Now, concerning (1.) assume that $X_t$ is real-valued. This should mean that $$ \sum_{j=1}^n (C(\lambda_j)\sin(t\lambda_j)+D(\lambda_j)\cos(t\lambda_j))=0. $$ Is it possible to get from this that it holds if and only if $$ A(\lambda_j)=\overline{A(\lambda_{n-j})},~~\lambda_j=-\lambda_{n-j}\text{ for }j=1,...,n-1 $$ and $A(\lambda_n)$ real-valued, i.e. $A(\lambda_n)=C(\lambda_n)$?