Question

$\mathrm{f}\mathrm{A}+\mathrm{B}+\mathrm{C}=,$ then $\tan \left(\frac{\mathrm{A}}{2}\right)\tan \left(\frac{\mathrm{B}}{2}\right)+\tan \left(\frac{\mathrm{B}}{2}\right)\tan \left(\frac{\mathrm{C}}{2}\right)+\tan \left(\frac{\mathrm{C}}{2}\right)\tan \left(\frac{\mathrm{A}}{2}\right)$ is equal to

• (A) π/6
• (B) 3
• (C) 2
• (D) 1

Answer 1

The Correct Option is D

Explanation:
Given that, $A+B+C=\pi$$\therefore \tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}$$\Rightarrow \tan \frac{B}{2}(\tan \frac{A}{2}+\tan \frac{C}{2})+\tan \frac{C}{2}\tan \frac{A}{2}$$\Rightarrow \tan \frac{B}{2}\left(\frac{\sin \frac{A}{2}\cos \frac{C}{2}+\sin \frac{C}{2}\cos \frac{A}{2}}{\cos \frac{A}{2}\cos \frac{C}{2}}\right)$$$\begin{array}{rl}+\frac{\sin \frac{C}{2}\sin \frac{A}{2}}{\cos \frac{C}{2}\cos \frac{A}{2}}& \\ \cos \frac{A}{2}\cos \frac{C}{2}& \frac{C}{2}\sin \frac{A}{2}\end{array}$$$\Rightarrow \frac{\tan \left(\frac{B}{2}\right)\{\sin \left(\frac{A+C}{2}\right)\}+\sin \frac{C}{2}}{2}$$\Rightarrow \frac{\sin (B/2)+\sin (C/2)\sin (A/2)}{\cos (A/2)\cos (C/2)}$$\Rightarrow \frac{\cos (A+C)/2+\sin (C/2)\sin (A/2)}{\cos (A/2)\cos (C/2)}$$\Rightarrow \frac{\cos (A/2)\cos (C/2)-\sin (A/2)}{\sin (C/2)+\sin (C/2)\sin (A/2)}$$$\begin{array}{c}\cos (A/2)\cos (C/2)\\ =\frac{\cos (A/2)\cdot \cos (C/2)}{\cos (A/2)\cdot \cos (C/2)}=1\end{array}$$