Question

Number of false relations of the following
$\left(i\right)tan|ta{n}^{-1}x|=|x|\left(ii\right)cot|co{t}^{-1}x|=|x|\left(iii\right)ta{n}^{-1}|tanx|=|x|\left(iv\right)sin|si{n}^{-1}x|=|x|$ is:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

If $x<0,|x|=-x,ta{n}^{1}x<0$
So $|ta{n}^{-1}x|=-ta{n}^{-1}x$.
$\therefore tan|ta{n}^{-1}x|=tan(-ta{n}^{-1}x)=-tanta{n}^{-1}x=-x$
and for $x\le 0,tan|ta{n}^{-1}x|=tanta{n}^{-1}x=x$.
$\therefore tan|ta{n}^{-1}x|=|x|.\therefore$ (i) is correct.
Similarly, (ii) and (iv) are correct.
In (iii), Let $x=\frac{3\pi }{4},ta{n}^{-1}|tanx|=ta{n}^{-1}|-1|$
$=ta{n}^{-1}1=\frac{\pi }{4}\ne \left|\frac{3\pi }{4}\right|=\frac{3\pi }{4}$.
Hence, there is one false relation.