Mathematics - Trigonometric Equation Question with Solution | TestHub

$2\left(\cos \beta -\cos \alpha \right)+\cos \alpha \cos \beta -1=0$        .....(i)

Use   $\cos \beta =\frac{1-{\tan }^2\frac{\beta }{2}}{1+{\tan }^2\frac{\beta }{2}} \mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ cos}\alpha =\frac{1-{\tan }^2\frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}$ in (i)

We get ${\tan }^2\left(\frac{\alpha }{2}\right)=3{\tan }^2\left(\frac{\beta }{2}\right) \Rightarrow \tan \left(\frac{\alpha }{2}\right)+\sqrt{3}\tan \frac{\beta }{2}=0 \mathrm{o}\mathrm{r}\mathrm{ tan}\left(\frac{\alpha }{2}\right)- \sqrt{3}\tan \left(\frac{\beta }{2}\right)=0$