MCQ

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If $f(x) = \cos \left( {{{\tan }^{ - 1}}\left( {\sin \left( {{{\cos }^{ - 1}}x} \right)} \right)} \right) + \sin \left( {{{\cot }^{ - 1}}\left( {\cos \left( {{{\sin }^{ - 1}}x} \right)} \right)} \right)$ has range $\left[ {m,M),} \right.$ then number of solutions of the equation $\operatorname{sgn} \left( {\left| {x - 1} \right| - 2} \right) = \ln \left| {x - 2} \right|$ is (where sgn denotes signum function)

• A
  $m^2+1$
• B
  $m^2-M$
• C
  $M^2+1$
• ✓
  $m^2+M$