Trigonometrical Ratios and Identities 2 Question 2
2. If $\alpha+\beta+\gamma=2 \pi$, then
$(1979,2 M)$
(a) $\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}$
(b) $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}+\tan \frac{\beta}{2} \tan \frac{Y}{2}+\tan \frac{Y}{2} \tan \frac{\alpha}{2}=1$
(c) $\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=-\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}$
(d) None of the above
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Answer:
Correct Answer: 2. (a)
Solution:
- Since, $\quad \frac{\alpha}{2}+\frac{\beta}{2}=\pi-\frac{\gamma}{2}$
$$ \therefore \quad \tan \frac{\alpha}{2}+\frac{\beta}{2}=\tan \pi-\frac{\gamma}{2} $$
$$ \Rightarrow \quad \frac{\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}}{1-\tan \frac{\alpha}{2} \tan \frac{\beta}{2}}=-\tan \frac{\gamma}{2} $$
$$ \Rightarrow \tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2} $$
