Question

If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.

Answer 1

Given:
$$\sin \alpha \sin \beta -\cos \alpha \cos \beta +1=0$$
$$\Rightarrow -(\cos \alpha \cos \beta -\sin \alpha \sin \beta )+1=0$$
$$\Rightarrow -\cos (\alpha +\beta )+1=0$$
$$\Rightarrow \cos (\alpha +\beta )=1$$
$$\text{ Therefore, }\sin (\alpha +\beta )=0....(1)(\text{ Since }\sin \theta =\sqrt{1-{\cos }^2\theta })$$
Hence ,
$$1+\cot \alpha \tan \beta =1+\frac{\cos \alpha \sin \beta }{\sin \alpha \cos \beta }$$
$$=\frac{\sin \alpha \cos \beta +\cos \alpha \sin \beta }{\sin \alpha \cos \beta }$$
$$=\frac{\sin (\alpha +\beta )}{\sin \alpha \cos \beta }$$
$$=0...\{\text{ From eq }(1)\}$$
Hence proved .