Question

If $A+B+C=\pi$ then which of the following are true?
i. $\tan 3A+\tan 3B+\tan 3C=\tan 3A\tan 3B\tan 3C$

ii. $\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$.

• A. only 1st statement is true
• B. only 2nd statement is true
• C. both statements are true
• D. both statements are false

Answer 1

Answer: (C)

both statements are true

Given, $A+B+C=\pi$

(i) $3A+3B+3C=3\pi \Rightarrow 3A+3B=3\pi -3C$

$\Rightarrow \tan (3A+3B)=\tan (3\pi -3C)$

$\Rightarrow \frac{\tan 3A+\tan 3B}{1-\tan 3A+\tan 3B}=-\tan 3C$ .............. $\because$ $\tan (A+B)=\frac{\tan \left(A\right)+\tan \left(B\right)}{1-\tan A.\tan B}$

$\Rightarrow \tan 3A+\tan 3B=-\tan 3C+\tan 3A+\tan 3B\tan 3C$

$\Rightarrow \tan 3A+\tan 3B+\tan 3C=\tan 3A\tan 3B\tan 3C$
(ii) $\frac{A}{2}+\frac{B}{2}+\frac{C}{2}=\frac{\pi }{2}\Rightarrow \frac{A}{2}+\frac{B}{2}=\frac{\pi }{2}-\frac{C}{2}$

$\Rightarrow \cot (\frac{A}{2}+\frac{B}{2})=\cot (\frac{\pi }{2}-\frac{C}{2})$

$\Rightarrow \frac{-1+\cot \frac{A}{2}\cot \frac{B}{2}}{\cot \frac{A}{2}+\cot \frac{B}{2}}=\tan \frac{C}{2}$

$\Rightarrow \frac{-1+\cot \frac{A}{2}\cot \frac{B}{2}}{\cot \frac{A}{2}+\cot \frac{B}{2}}=\frac{1}{\cot \frac{C}{2}}$

$\Rightarrow -\cot \frac{C}{2}+\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}=\cot \frac{A}{2}+\cot \frac{B}{2}$

$\Rightarrow \cot \frac{C}{2}+\cot \frac{A}{2}+\cot \frac{B}{2}=\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$