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text  If  end text f left parenthesis x right parenthesis equals fraction numerator cos x sin x over denominator tan x plus cot x end fraction text end textthen function f(x) is

  1. Even
  2. Odd
  3. Both even and odd
  4. Neither odd nor even

The correct answer is: Even


    f left parenthesis negative x right parenthesis equals fraction numerator cos left parenthesis negative x right parenthesis sin left parenthesis negative x right parenthesis over denominator tan left parenthesis negative x right parenthesis plus cot left parenthesis negative x right parenthesis end fraction equals fraction numerator cos x sin x over denominator tan x plus cot x end fraction equals f left parenthesis x right parenthesis not stretchy rightwards double arrow f left parenthesis x right parenthesis text  is even  end text

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    text  Let  end text g colon R not stretchy rightwards arrow open parentheses 0 comma pi over 3 close square brackets text  be defined by  end text g left parenthesis x right parenthesis equals cos to the power of negative 1 end exponent open parentheses fraction numerator x squared minus k over denominator 1 plus x squared end fraction close parentheses spaceThen find the possible values of  k for which g is surjective function

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    text  If  end text f colon N not stretchy rightwards arrow Z space f left parenthesis n right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1em end attributes row cell fraction numerator n minus 1 over denominator 2 end fraction comma text  when  end text n text  is odd  end text end cell row cell negative n over 2 comma text  when  end text n text  is even  end text end cell end table close
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    text  If  end text f colon N not stretchy rightwards arrow Z space f left parenthesis n right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1em end attributes row cell fraction numerator n minus 1 over denominator 2 end fraction comma text  when  end text n text  is odd  end text end cell row cell negative n over 2 comma text  when  end text n text  is even  end text end cell end table close
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