Question
The set of values of $$x$$ for which the identity $${\cos ^{ - 1}}x + {\cos ^{ - 1}}\left( {\frac{x}{2} + \frac{1}{2}\sqrt {3 - 3{x^2}} } \right) = \frac{\pi }{3}$$ holds good is
A.
$$\left[ {0,1} \right]$$
B.
$$\left[ {0,\frac{1}{2}} \right]$$
C.
$$\left[ {\frac{1}{2} , 1} \right]$$
D.
$$\left\{ { - 1,0,1} \right\}$$
Answer :
$$\left[ {\frac{1}{2} , 1} \right]$$
Solution :
$$\eqalign{
& {\bf{Case 1 :}} \cr
& {\text{If }}0 \leqslant x \leqslant \frac{1}{2},\,\,{\text{then }}{\cos ^{ - 1}}\left( {\frac{x}{2} + \frac{1}{2}\sqrt {3 - 3{x^2}} } \right) \cr
& {\cos ^{ - 1}}\left( {x \times \frac{1}{2} + \sqrt {1 - {x^2}} \frac{{\sqrt 3 }}{2}} \right) \cr
& = {\cos ^{ - 1}}x - {\cos ^{ - 1}}\frac{1}{2} \cr} $$
Therefore, the equation is
$$\eqalign{
& {\cos ^{ - 1}}x + {\cos ^{ - 1}}x - {\cos ^{ - 1}}\frac{1}{2} = \frac{\pi }{3} \cr
& \Rightarrow x = \frac{1}{2}. \cr
& {\bf{Case 2 :}} \cr
& {\text{If }}\frac{1}{2} \leqslant x \leqslant 1,{\text{ then}} \cr
& {\cos ^{ - 1}}\left( {\frac{x}{2} + \frac{1}{2}\sqrt {3 - 3{x^2}} } \right) = {\cos ^{ - 1}}\frac{1}{2} - {\cos ^{ - 1}}x \cr} $$
Therefore, the equation is
$${\cos ^{ - 1}}x + {\cos ^{ - 1}}\frac{1}{2} - {\cos ^{ - 1}}x = \frac{\pi }{3},$$
which is an identity.
Hence, the identity holds good for $$x \in \left[ {\frac{1}{2},1} \right].$$