Question

If $\sin \alpha sin\beta -cos\alpha cos\beta +1=0$ then show that $1+\cot \alpha \tan \beta =0$.

Answer 1

Given $\sin \alpha \sin \beta -\cos \alpha \cos \beta +1=0$

$⟹\cos \alpha \cos \beta -\sin \alpha \sin \beta =1$

$⟹\cos (\alpha +\beta )=\cos 2n\pi$ where $n$ is an integer

$⟹\alpha +\beta =2n\pi$

$⟹\alpha =2n\pi -\beta$

$1+\cot \alpha \tan \beta =1+\cot (2n\pi -\beta )\tan \beta =1-\cot \beta \tan \beta =1-1=0$