There are $d$ real unknowns $x_1,x_2,\ldots,x_d$ satisfy the following equations $$\frac{x_1}{\sqrt{x_1^2+x_k^2}}=\cos \theta_k,\frac{x_k}{\sqrt{x_1^2+x_k^2}}=\sin \theta_k,\forall k\in\{2,3,\ldots,d\}$$ and $$x_1^2+x_2^2+\cdots+x_d^2=1,$$ where $\theta_k$'s are known constants.
I guess that the number of solutions should be greater than $1$ but have no idea how to prove it. Thank you for your help!
From your first condition we have $$ x_k = x_1 \tan \theta_k $$ and from second we have $$ x_1^2 = \frac{1}{1 + \sum_{k=2}^d \tan\theta_k^2} $$ There are two choices for $x_1$ and each choice gives unique value for $x_k$, $k\geq2$.