Question

Assertion :If $a>b>0$ & $ac<-1<bc$, then ${\tan }^{-1}\left(\frac{a-b}{1+ab}\right)+{\tan }^{-1}\left(\frac{b-c}{1+bc}\right)+{\tan }^{-1}\left(\frac{c-a}{1+ac}\right)=\pi$ Reason: ${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right)\forall x,y$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

${\tan }^{-1}x-{\tan }^{-1}y$
$\{\begin{array}{c}{\tan }^{-1}\left(\frac{x-y}{1+xy}\right)ifxy>-1\\ \pi +{\tan }^{-1}\left(\frac{x-y}{1+xy}\right)ifx>0,y<0,xy<-1\\ -\pi +{\tan }^{-1}\left(\frac{x-y}{1+xy}\right)ifx<0,y>0,xy<-1\end{array}$

Now $a>b>0soab>0$

$\therefore {\tan }^{-1}\left(\frac{a-b}{1+ab}\right)={\tan }^{-1}a-{\tan }^{-1}b$ .....( i )

And $bc>-1$

$\therefore {\tan }^{-1}\left(\frac{b-c}{1+bc}\right)={\tan }^{-1}b-{\tan }^{-1}c$......( ii )

Also $ac<-1$ & $a>0,c<0$

$\therefore {\tan }^{-1}\left(\frac{c-a}{1+ca}\right)=\pi +{\tan }^{-1}c-{\tan }^{-1}a$.....( iii )

On adding (i), (ii) & (iii) we get

${\tan }^{-1}\left(\frac{a-b}{1+ab}\right)+{\tan }^{-1}\left(\frac{b-c}{1+bc}\right)+{\tan }^{-1}\left(\frac{c-a}{1+ca}\right)$

$\therefore$ Assertion (A) is correct

$\therefore$ By fact Reason is false