Question

If $\cos (\alpha +\beta )\sin \left(\gamma +\theta \right)=\cos \left(\alpha -\beta \right)\sin \left(\gamma -\theta \right)$, Find $\tan \theta -\tan \alpha \tan \beta \tan \gamma$

Answer 1

Consider the given equation.

$\cos (\alpha +\beta ).\sin (\gamma +\theta )=\cos (\alpha -\beta ).\sin (\gamma -\theta )$

$(\cos \alpha \cos \beta -\sin \alpha \sin \beta )(\sin \gamma \cos \theta +\sin \theta \cos \gamma )=(\cos \alpha \cos \beta +\sin \alpha \sin \beta )(\sin \gamma \cos \theta -\cos \gamma \sin \theta )$

$2\cos \alpha \cos \beta \sin \theta \cos \gamma =2\sin \alpha \sin \beta \sin \gamma \cos \theta$

$\tan \theta =\tan \alpha tan\beta \tan \gamma$

$\tan \theta -\tan \alpha tan\beta \tan \gamma =0$

Hence, the value is $0$.