If $\theta$₁ and $\theta$₂ are opparent angles of dip.

Let $\alpha$ be the angle which one of the plane make with the magnetic meridian.

For magnetic meridian and plane (1)

tan $\theta$₁ = ${{\tan \theta } \over {\cos \alpha }}$

$\Rightarrow$ cos $\alpha$ = ${{\tan \theta } \over {\tan {\theta _1}}}$ .....(1)

For magnetic meridian and plane (2)

tan $\theta$₁ = ${{\tan \theta } \over {\cos \left( {90^\circ - \alpha } \right)}}$

$\Rightarrow$ $\sin \alpha = {{\tan \theta } \over {\tan {\theta _2}}}$ ....(2)

(1)² + (2)²

1 = ${{{{\tan }^2}\theta } \over {{{\tan }^2}{\theta _1}}} + {{{{\tan }^2}\theta } \over {{{\tan }^2}{\theta _2}}}$

$\therefore$ cot²$\theta$ = cot²$\theta$₁ + cot²$\theta$₂