Question

If $\alpha +\beta +\gamma =2\pi$, then

• 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. $\tan \frac{\alpha }{2}\tan \frac{\beta }{2}+\tan \frac{\beta }{2}\tan \frac{\gamma }{2}+\tan \frac{\gamma }{2}\tan \frac{\alpha }{2}=1$
• 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 these

Answer 1

The correct option is A

$\tan \frac{\alpha }{2}+\tan \frac{\beta }{2}+\tan \frac{\gamma }{2}=\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\tan \frac{\gamma }{2}$

Explanation for the correct option:

Given, $\alpha +\beta +\gamma =2\pi$

$\Rightarrow$ $\frac{(\alpha +\beta +\gamma )}{2}=\pi$

Take “tan” on both sides,

$\tan \left(\frac{\alpha }{2}+\frac{\beta }{2}+\frac{\gamma }{2}\right)=\tan \pi$

$\Rightarrow$$\frac{\tan {\displaystyle \frac{\alpha }{2}}+\tan {\displaystyle \frac{\beta }{2}}+\tan {\displaystyle \frac{\gamma }{2}}–\tan {\displaystyle \frac{\alpha }{2}}\tan {\displaystyle \frac{\beta }{2}}\tan {\displaystyle \frac{\gamma }{2}}}{1–\tan {\displaystyle \frac{\alpha }{2}}\tan {\displaystyle \frac{\beta }{2}}–\tan {\displaystyle \frac{\beta }{2}}\tan {\displaystyle \frac{\gamma }{2}}–\tan {\displaystyle \frac{\gamma }{2}}\tan {\displaystyle \frac{\alpha }{2}}}=0$

$\Rightarrow$ $\tan \frac{\alpha }{2}+\tan \frac{\beta }{2}+\tan \frac{\gamma }{2}–\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\tan \frac{\gamma }{2}=0$

$\mathbf{\therefore }\mathit{t}\mathit{a}\mathit{n}\mathbf{}\frac{\mathbf{\alpha }}{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\frac{\mathbf{\beta }}{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\frac{\mathbf{\gamma }}{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\frac{\mathbf{\alpha }}{\mathbf{2}}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\frac{\mathbf{\beta }}{\mathbf{2}}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\frac{\mathbf{\gamma }}{\mathbf{2}}$

Hence, Option ‘A’ is Correct.