# If the projections of $OA$ and $OB$ on the plane $z=0$ make angles $\phi_1$ and $\phi_2$,respectively,with the $x-$axis

$OA,OB,OC,$ with $O$ as origin, are three mutually perpendicular lines whose direction cosines are $l_1,m_1,n_1;l_2,m_2,n_2$ and $l_3,m_3,n_3$ respectively. If the projections of $OA$ and $OB$ on the plane $z=0$ make angles $\phi_1$ and $\phi_2$, respectively, with the $X-$axis, prove that $\tan(\phi_1-\phi_2)=\pm\dfrac{n_3}{n_1n_2}$

The given conditions in your question are sufficient to show that the matrix $$A=\begin{pmatrix} l_1&m_1&n_1\\l_2&m_2&n_2\\l_3&m_3&n_3 \end{pmatrix}$$ is an orthogonal matrix. It should be obvious now that $n_1^2+n_2^2+n_3^2=1$