Question

If $\cos (\alpha +\beta )\sin (\gamma +\delta )=\cos (\alpha -\beta )\sin (\gamma +\delta )$, then prove that $\cot \alpha \cot \beta \cot \gamma =\cot \delta$

Answer 1

From the given relation, we have
$\frac{\sin (\gamma +\delta )}{\sin (\gamma -\delta )}=\frac{\cos (\alpha -\beta )}{\cos (\alpha +\beta )}$
Now apply componendo and dividendo
$\therefore \frac{\sin (\gamma +\delta )+\sin (\gamma -\delta )}{\sin (\gamma +\delta )-\sin (\gamma -\delta )}=\frac{\cos (\alpha -\beta )+\cos (\alpha +\beta )}{\cos (\alpha -\beta )-\cos (\alpha +\beta )}$
or $\therefore \cot \delta =\cot \alpha \cot \beta \cot \gamma$