$k\cos\beta \sin\alpha-\sin\beta+d\sqrt{1+k^2}=0$ ，Find $\beta$ $k\cos\beta \sin\alpha-\sin\beta+d\sqrt{1+k^2}=0$. Find $\beta$.
$K$ and $d$ are constant.
$d∈[0,\pm1],k∈[0,\pm\infty)$.
This is an equation in a spherical coordinate system.  Where $\alpha$ is longitude and $\beta$ is latitude.

\begin{align}
& A\cos\beta + B\sin\beta \\[10pt]
= {} & \sqrt{A^2+B^2} \left( \frac A{\sqrt{A^2+B^s}} \cos\beta + \frac B {\sqrt{A^2+B^2}} \sin\beta \right) \\[10pt]
= {} & \sqrt{A^2+B^2} \Big( \sin\gamma \cos\beta + \cos\gamma\sin\beta\Big) \\
& \quad \text{where } \tan\gamma = A/B \\[10pt]
= {} & \sqrt{A^2+B^2} \, \sin(\beta+\gamma)
\end{align}
You have $A=k\cos\alpha$ and $B=-1.$
