Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $C o s^{- 1} x - 2 S i n^{- 1} x = C o s^{- 1} 2 x$ is equal to

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$0$

$1$

$\frac{1}{2}$

$- \frac{1}{2}$

answer is A.

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Detailed Solution

$\begin{array}{l} C o s^{- 1} x - 2 S i n^{- 1} x = C o s^{- 1} 2 x \Rightarrow \frac{π}{2} - 3 S i n^{- 1} x = \frac{π}{2} - S i n^{- 1} 2 x \\ \Rightarrow 3 S i n^{- 1} x = S i n^{- 1} 2 x \end{array}$

$\begin{array}{l} \Rightarrow S i n^{- 1} (3 x - 4 x^{3}) = S i n^{- 1} 2 x \Rightarrow 3 x - 4 x^{3} = 2 x \Rightarrow 4 x^{3} - x = 0 \\ \Rightarrow S u m o f x v a l u e s = 0 \end{array}$