text: • # question_answer Statement-1: If$1/2\le x\le 1$, then ${{\cos }^{-1}}x+{{\cos }^{-1}}\left[ \frac{x}{2}+\frac{\sqrt{3-3{{x}^{2}}}}{2} \right]$is equal to$\pi /3$ Statement 2:${{\sin }^{-1}}\left( 2x\sqrt{1-{{x}^{2}}} \right)=2{{\sin }^{-1}}x,$if$x\in \left[ -1/\sqrt{2},\,\,1/\sqrt{2} \right]$ A)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.B)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.C)  Statement-1 is true, Statement-2 is false.D)  Statement-1 is false, Statement-2 is true.
  
  In statement $1$ put $x=\cos \theta$ then$0\le \theta \le \pi /3$ $L.H.S={{\cos }^{-1}}(\cos \theta )+{{\cos }^{-1}}\left[ \frac{1}{2}\cos \theta +\frac{\sqrt{3}}{2}\sin \theta \right]$$=\theta +{{\cos }^{-1}}(\cos (\pi /3-\theta ))=\theta +\pi /3-\theta =\frac{\pi }{3}$so statement $1$ is true. In statement $2$, put $x=\sin \theta$ then$-\pi /4\le \theta \le \pi /4$ so${{\sin }^{-1}}(2x\sqrt{1-{{x}^{2}}})={{\sin }^{-1}}(2\sin \theta \cos \theta )$ $={{\sin }^{-1}}(\sin 2\theta )=2\theta =2{{\sin }^{-1}}x$
explanation: The document contains a math problem with two statements and a multiple-choice question. It also includes a detailed solution for each statement, verifying the truth of the statements.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: • # question_answer Statement-1: If$1/2\le x\le 1$, then ${{\cos }^{-1}}x+{{\cos }^{-1}}\left[ \frac{x}{2}+\frac{\sqrt{3-3{{x}^{2}}}}{2} \right]$is equal to$\pi /3$ Statement 2:${{\sin }^{-1}}\left( 2x\sqrt{1-{{x}^{2}}} \right)=2{{\sin }^{-1}}x,$if$x\in \left[ -1/\sqrt{2},\,\,1/\sqrt{2} \right]$ A)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.B)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.C)  Statement-1 is true, Statement-2 is false.D)  Statement-1 is false, Statement-2 is true.
  
  In statement $1$ put $x=\cos \theta$ then$0\le \theta \le \pi /3$ $L.H.S={{\cos }^{-1}}(\cos \theta )+{{\cos }^{-1}}\left[ \frac{1}{2}\cos \theta +\frac{\sqrt{3}}{2}\sin \theta \right]$$=\theta +{{\cos }^{-1}}(\cos (\pi /3-\theta ))=\theta +\pi /3-\theta =\frac{\pi }{3}$so statement $1$ is true. In statement $2$, put $x=\sin \theta$ then$-\pi /4\le \theta \le \pi /4$ so${{\sin }^{-1}}(2x\sqrt{1-{{x}^{2}}})={{\sin }^{-1}}(2\sin \theta \cos \theta )$ $={{\sin }^{-1}}(\sin 2\theta )=2\theta =2{{\sin }^{-1}}x$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem with two statements and a multiple-choice question. It also includes a detailed solution for each statement, verifying the truth of the statements. 
  
  Label: question-and-answer