Let \( \alpha \) and \( \beta \) be non-zero real numbers such that...

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Let \( \alpha \) and \( \beta \) be non-zero real numbers such that \( 2(\cos \beta-\cos \alpha)+\cos \alpha \cos \beta=1 \). Then which of the
\( \mathrm{P} \) following is/are true?
(A) \( \sqrt{3} \tan \left(\frac{\alpha}{2}\right)+\tan \left(\frac{\beta}{2}\right)=0 \)
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(B) \( \sqrt{3} \tan \left(\frac{\alpha}{2}\right)-\tan \left(\frac{\beta}{2}\right)=0 \)
(C) \( \tan \left(\frac{\alpha}{2}\right)+\sqrt{3} \tan \left(\frac{\beta}{2}\right)=0 \)
(D) \( \tan \left(\frac{\alpha}{2}\right)-\sqrt{3} \tan \left(\frac{\beta}{2}\right)=0 \)
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