Discussion Forum

Let α and β be non zero real numbers such $2(\cos\beta-\cos\alpha)+\cos\alpha\cos\beta=1$ . Then which of the following is/are true?

Answer:

Explanation:

We have

$2(\cos\beta-\cos\alpha)+\cos\alpha\cos\beta=1$

 or    $4(\cos\beta-\cos\alpha)+2\cos\alpha\cos\beta=2$

$\Rightarrow$   $1-\cos\alpha+\cos\beta-\cos\alpha\cos\beta$

$=3+3\cos\alpha-3\cos\beta-3\cos\alpha\cos\beta$

$\Rightarrow$   $(1-\cos\alpha)(1+\cos\beta)$

                     =$3(1-\cos\alpha)(1+\cos\beta)$

$\Rightarrow$         $\frac{(1-\cos\alpha)}{(1+\cos\alpha)}=\frac{3(1-\cos\beta)}{(1+\cos\beta}$

$\Rightarrow$   $\tan^{2}\frac{\alpha}{2}=3\tan^{2}\frac{\beta}{2}$

$\therefore$     $\tan\frac{\alpha}{2}\pm \sqrt{3}\tan\frac{\beta}{2}=0$