Nếu (tan frac{beta }{2} = 4tan frac{alpha }{2}) thì (tan frac{{beta  – alpha }}{2}) bằng :

Câu hỏi:

Nếu \(\tan \frac{\beta }{2} = 4\tan \frac{\alpha }{2}\) thì \(\tan \frac{{\beta  – \alpha }}{2}\) bằng :

A.
\(\frac{{3\sin \alpha }}{{5 – 3\cos \alpha }}.\)

B.
\(\frac{{3\sin \alpha }}{{5 + 3\cos \alpha }}.\)

C.
\(\frac{{3\cos \alpha }}{{5 – 3\cos \alpha }}.\)

D.
\(\frac{{3\cos \alpha }}{{5 + 3\cos \alpha }}.\)

Đáp án đúng: A

Ta có \(\tan \frac{{\beta  – \alpha }}{2} = \frac{{\tan \frac{\beta }{2} – \tan \frac{\alpha }{2}}}{{1 + \tan \frac{\beta }{2}.\tan \frac{\alpha }{2}}} = \frac{{3\tan \frac{\alpha }{2}}}{{1 + 4{{\tan }^2}\frac{\alpha }{2}}} = \frac{{3\sin \frac{\alpha }{2}.\cos \frac{\alpha }{2}}}{{1 + 3{{\sin }^2}\frac{\alpha }{2}}} = \frac{{3\sin \alpha }}{{5 – 3\cos \alpha }}.\)