Question

${\cos }^{-1}x+{\cos }^{-1}(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^2})$ is equal to

• A. $\frac{\pi }{3}$ for $xϵ[\frac{1}{2},1]$
• B. $\frac{\pi }{3}$ for $xϵ[0,\frac{1}{2}]$
• C. $2{\cos }^{-1}x-{\cos }^{-1}\frac{1}{2}$ for $xϵ[\frac{1}{2},1]$
• D. $2{\cos }^{-1}x-{\cos }^{-1}\frac{1}{2}$ for $xϵ[0,\frac{1}{2}]$

Answer 1

Answer: (A), (D)

The correct options are
A $\frac{\pi }{3}$ for $xϵ[\frac{1}{2},1]$
D $2{\cos }^{-1}x-{\cos }^{-1}\frac{1}{2}$ for $xϵ[0,\frac{1}{2}]$

$co{s}^{-1}x+co{s}^{-1}(x.\frac{1}{2}+\frac{\sqrt{3}}{2}\sqrt{1-x^2})=co{s}^{-1}x+co{s}^{-1}\frac{1}{3}-co{s}^{-1}x$
For $\frac{1}{2}\le x\le 1$
$co{s}^{-1}x+co{s}^{-1}(x.\frac{1}{2}+\frac{\sqrt{3}}{2}\sqrt{1-x^2})=co{s}^{-1}x+\frac{\pi }{3}-co{s}^{-1}x=\frac{\pi }{3}$
And for $0\le x\le \frac{1}{2}$
$o{s}^{-1}x+co{s}^{-1}(x.\frac{1}{2}+\frac{\sqrt{3}}{2}\sqrt{1-x^2})=co{s}^{-1}x+co{s}^{-1}x-co{s}^{-1}\frac{1}{2}=2co{s}^{-1}x-co{s}^{-1}\frac{1}{2}$