If $\theta$ be the true angle of dip and ${\theta }_1 and {\theta }_2$ are the apparent dips observed in two vertical planes at right angles to each other, then the relation is :

Let $B_H$ and  $B_V$  be the horizonal and vertical components of earth's magnetic field $\overrightarrow{\beta }$. Since $\theta$ is the angle of dip

$\therefore \mathrm{tan\theta }=\frac{{\mathrm{B}}_{\mathrm{V}}}{{\mathrm{B}}_{\mathrm{H}}} \mathrm{or} \mathrm{cot\theta }=\frac{{\mathrm{B}}_{\mathrm{H}}}{{\mathrm{B}}_{\mathrm{V}}}\dots (\mathrm{i})$

Suppose planes 1 and 2 are two mutually perpendicular planes and respectively make angles $\theta$ and 90°-$\theta$ with the magnetic meridian. The vertical components of earth's magnetic field remain same in the two planes but the effective horizontal components in the planes will be

$$\begin{gathered} {\mathrm{B}}_1={\mathrm{B}}_{\mathrm{H}}\mathrm{cos\theta } \\ \text{ and }{\mathrm{B}}_2={\mathrm{B}}_{\mathrm{H}}\mathrm{sin\theta } \\ \text{ The angles of }dip{\mathrm{\theta }}_1\text{ and }{\mathrm{\theta }}_2\text{ in the two planes are given by} \\ \begin{array}{l}{\mathrm{tan\theta }}_1=\frac{{\mathrm{B}}_{\mathrm{V}}}{{\mathrm{B}}_1}\\ {\mathrm{tan\theta }}_1=\frac{{\mathrm{B}}_{\mathrm{V}}}{{\mathrm{B}}_{\mathrm{H}}\mathrm{cos\theta }}\\ \text{ or }{\mathrm{cot\theta }}_1=\frac{{\mathrm{B}}_{\mathrm{H}}\mathrm{cos\theta }}{{\mathrm{B}}_{\mathrm{V}}} . . . \left(\mathrm{ii}\right)\end{array} \\ \mathrm{Similarly}, \\ {\mathrm{cot\theta }}_2=\frac{{\mathrm{B}}_{\mathrm{H}}\mathrm{sin\theta }}{{\mathrm{B}}_{\mathrm{V}}} . . . .\left(\mathrm{iii}\right) \\ \text{ From eqns. (ii) and (iii) } \\ {\cot }^2{\mathrm{\theta }}_1+{\cot }^2{\mathrm{\theta }}_2=\frac{{\mathrm{B}}_{\mathrm{H}}^2}{{\mathrm{B}}_{\mathrm{V}}^2}\left({\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }\right)=\frac{{\mathrm{B}}_{\mathrm{H}}^2}{{\mathrm{B}}_{\mathrm{V}}^2} \\ \therefore {\cot }^2{\mathrm{\theta }}_1+{\cot }^2{\mathrm{\theta }}_2={\cot }^2\mathrm{\theta } [ \mathrm{from} \mathrm{eqn}. (\mathrm{i}\left)\right] \end{gathered}$$