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Question

Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals

  1. 1 over 8
  2. 0
  3. 1 over 32
  4. straight infinity

Hint:

In this question firstly we will solve the trigonometric equation and after eliminating the tan part. Then we will use the L'hospital theorem twice and later evaluate and put the value.

The correct answer is: 1 over 32


    space space space space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction
equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets tan begin display style straight pi over 4 end style minus tan space x over 2 close square brackets over denominator open square brackets tan space begin display style straight pi over 4 end style plus tan space x over 2 close square brackets end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction
equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space tan space open parentheses straight pi over 4 minus x over 2 close parentheses. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction
equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator tan space open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator 4 left square bracket pi minus 2 x right square bracket squared end fraction
equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket squared end fraction
U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space
equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative cos space x over denominator 2 left parenthesis straight pi minus 2 straight x right parenthesis. left parenthesis negative 2 right parenthesis end fraction
equals 1 over 16 space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator left parenthesis straight pi minus 2 straight x right parenthesis end fraction
equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction
equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction
U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space
equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative sin space x over denominator negative 2 end fraction
equals 1 over 32 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space sin space x
equals 1 over 32 cross times sin space straight pi over 2
equals 1 over 32 space cross times 1
equals 1 over 32.

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