from a point on the ground, the angles of elevation of a bird flying with a constant speed in a horizontal direction at equal intervals of time are $\alpha, \beta, \gamma$ and $\delta$. THEN $\cot^{ 2 }\alpha - \cot^{ 2 }\delta=3(\cot^{ 2 }\beta-\cot^{ 2 }\gamma)$;

Assume the bird flies at height h with constant speed v, making horizontal distances from the observer proportional to time. The distance at interval n is nv, so the tangent of the angle is h divided by nv, meaning the cotangent is nv divided by h. The left side becomes 1v squared divided by h squared minus 4v squared divided by h squared, which is minus 3v squared divided by h squared. The right side is 3 multiplied by 2v squared divided by h squared minus 3v squared divided by h squared, which also equals minus 3v squared divided by h squared, proving the statement is true.

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