Question

For $xϵ[0,\frac{1}{2}],f\left(x\right)=\tan ({\sin }^{-1}(\frac{x}{\sqrt{2}}+\sqrt{\frac{1-x^2}{2}})-{\sin }^{-1}x)$ is

• A. discontinuous at one point
• B. discontinuous at two point
• C. continuous throughout
• D. discontinuous at infinite points

Answer 1

The correct option is C

continuous throughout

${\sin }^{-1}(\frac{x}{\sqrt{2}}+\sqrt{\frac{1-x^2}{2}})={\sin }^{-1}(x\sqrt{1-{\left(\frac{1}{\sqrt{2}}\right)}^2}+\frac{1}{\sqrt{2}}\sqrt{1-x^2})$

$={\sin }^{-1}\frac{1}{\sqrt{2}}$

$\Rightarrow f\left(x\right)=\tan \left({\sin }^{-1}\frac{1}{\sqrt{2}}\right)=\tan \frac{\pi }{4}$

So, $f\left(x\right)=1$

Hence, the function is continuous throughout.