Maths-
General
Easy

Question

If tan space alpha equals fraction numerator 1 over denominator square root of x open parentheses x squared plus x plus 1 close parentheses end root end fraction comma tan space beta equals fraction numerator square root of x over denominator square root of x squared plus x plus 1 end root end fraction and tan space gamma equals square root of x to the power of negative 3 end exponent plus x to the power of negative 2 end exponent plus x to the power of negative 1 end exponent end root then alpha plus beta =

  1. gamma
  2. 2gamma
  3. negative gamma
  4. gamma divided by 2

hintHint:

In this question, we have to find the value of alpha plus beta, if tan space alpha equals fraction numerator 1 over denominator square root of x open parentheses x squared plus x plus 1 close parentheses end root end fraction comma tan space beta equals fraction numerator square root of x over denominator square root of x squared plus x plus 1 end root end fraction and tan space gamma equals square root of x to the power of negative 3 end exponent plus x to the power of negative 2 end exponent plus x to the power of negative 1 end exponent end root. First we will find the value of tan(alpha plus beta) and later equate it with the answer to find the value of alpha plus beta.

The correct answer is: gamma


    tan space alpha equals fraction numerator 1 over denominator square root of x open parentheses x squared plus x plus 1 close parentheses end root end fraction comma space tan space beta equals fraction numerator square root of x over denominator square root of x squared plus x plus 1 end root end fraction space a n d space tan space gamma equals square root of x to the power of negative 3 end exponent plus x to the power of negative 2 end exponent plus x to the power of negative 1 end exponent end root
space space space space tan space left parenthesis alpha plus beta right parenthesis
equals fraction numerator tan space alpha plus tan space beta over denominator 1 minus tan space alpha. tan space beta end fraction
equals fraction numerator fraction numerator 1 over denominator square root of x open parentheses x squared plus x plus 1 close parentheses end root end fraction plus fraction numerator square root of x over denominator square root of x squared plus x plus 1 end root end fraction over denominator 1 minus fraction numerator 1 over denominator square root of x open parentheses x squared plus x plus 1 close parentheses end root end fraction. fraction numerator square root of x over denominator square root of x squared plus x plus 1 end root end fraction end fraction
equals fraction numerator square root of x squared plus x plus 1 end root plus x square root of x squared plus x plus 1 end root over denominator square root of x left parenthesis x squared plus x plus 1 right parenthesis minus square root of x end fraction
equals fraction numerator open parentheses x plus 1 close parentheses square root of x squared plus x plus 1 end root over denominator square root of x open parentheses x squared plus x plus 1 minus 1 close parentheses end fraction
equals fraction numerator left parenthesis x plus 1 right parenthesis square root of x squared plus x plus 1 end root over denominator x square root of x open parentheses x plus 1 close parentheses end fraction
equals fraction numerator square root of x squared plus x plus 1 end root over denominator square root of x cubed end root end fraction
equals square root of fraction numerator x squared plus x plus 1 over denominator x cubed end fraction end root
equals square root of 1 over x plus 1 over x squared plus 1 over x cubed end root
equals square root of x to the power of negative 3 end exponent plus x to the power of negative 2 end exponent plus x to the power of negative 1 end exponent end root
equals tan space gamma
A s comma space tan space left parenthesis alpha plus beta right parenthesis equals tan space gamma
rightwards double arrow alpha plus beta equals gamma

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