Question

If $a\sin (\theta +\alpha )=b\sin (\theta +\beta )$, then $\cot \theta =\frac{a\cos \alpha -b\cos \beta }{b\sin \beta -a\sin \alpha }$.

• A. True
• B. False

Answer 1

The correct option is A True

$a\sin (\theta +\alpha )=b\sin (\theta +\beta )$

$a(\sin \theta \cos \alpha +\cos \theta \sin \alpha )=b(\sin \theta \cos \beta +\cos \theta \sin \beta )$

$a\sin \theta \cos \alpha +a\cos \theta \sin \alpha =b\sin \theta \cos \beta +b\cos \theta \sin \beta$

$a\sin \theta \cos \alpha -b\sin \theta \cos \beta =b\cos \theta \sin \beta -a\cos \theta \sin \alpha$

$\sin \theta \left(a\cos \alpha -b\cos \beta \right)=\cos \theta \left(b\sin \beta -a\sin \alpha \right)$

$\cot \theta =\frac{a\cos \alpha -b\cos \beta }{b\sin \beta -a\sin \alpha }$

prove.