Question
The solution set of the equation $${\cos ^{ - 1}}x - {\sin ^{ - 1}}x = {\sin ^{ - 1}}\left( {1 - x} \right)$$ is
A.
$$\left[ { - 1,1} \right]$$
B.
$$\left[ {0,\frac{1}{2}} \right]$$
C.
$$\left[ { - 1,0} \right]$$
D.
None of these
Answer :
None of these
Solution :
$${\sin ^{ - 1}}x,{\cos ^{ - 1}}x\,$$ exist for $$ - 1 \leqslant x \leqslant 1.$$ But for $${\sin ^{ - 1}}\left( {1 - x} \right)$$ we must have $$ - 1 \leqslant 1 - x \leqslant 1,\,{\text{i}}{\text{.e}}{\text{., }}0 \leqslant x \leqslant 2.$$ So the equation may hold for $$0 \leqslant x \leqslant 1.$$ Therefore, the options $$A, B$$ and $$C$$ are incorrect.