Question

Solution of equation $tan\left(co{s}^{-1}x\right)=sin\left[co{t}^{-1}\frac{1}{2}\right]$ is

• A.
• B.
  
• C.
  
• D. None of these

Answer 1

The correct option is B

We have, $tan\left(co{s}^{-1}x\right)=sin\left[co{t}^{-1}\frac{1}{2}\right]$
$\Rightarrow tan\left[ta{n}^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right]=sin\left(ta{n}^{-1}2\right)$
$\Rightarrow tan\left[ta{n}^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right]=sin\left(si{n}^{-1}\frac{2}{\sqrt{1+4}}\right)$
$\Rightarrow tan\left[ta{n}^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right]=sin\left(si{n}^{-1}\frac{2}{\sqrt{5}}\right)$
$\Rightarrow \frac{\sqrt{1-x^2}}{x}=\frac{2}{\sqrt{5}}\Rightarrow 5(1-x^2)=4x^2$
$\Rightarrow 9x^2=5,\therefore x=\pm \left[\frac{\sqrt{5}}{3}\right]$