Maths-
General
Easy

Question

3 tan to the power of 6 space 10 to the power of ring operator minus 27 tan to the power of 4 space 10 to the power of ring operator plus 33 tan squared space 10 to the power of ring operator space=

  1. 0
  2. 1
  3. 2
  4. 3

Hint:

In this question, we have too find the value of the given equation. For this we will use the value of tan 30 and later simplify it in the required way to find the value.

The correct answer is: 1


    3 tan to the power of 6 space 10 to the power of ring operator minus 27 tan to the power of 4 space 10 to the power of ring operator plus 33 tan squared space 10 to the power of ring operator
A s comma space tan space 30 degree equals fraction numerator 1 over denominator square root of 3 end fraction
rightwards double arrow tan 3 left parenthesis 10 degree right parenthesis equals fraction numerator 1 over denominator square root of 3 end fraction
space rightwards double arrow fraction numerator left parenthesis 3 tan 10 degree minus tan cubed 10 degree right parenthesis over denominator left parenthesis 1 minus 3 tan squared 10 degree right parenthesis end fraction equals fraction numerator 1 over denominator square root of 3 end fraction
S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon
rightwards double arrow left parenthesis 3 tan 10 degree minus tan cubed 10 degree right parenthesis squared over left parenthesis 1 minus 3 tan squared 10 degree right parenthesis squared equals 1 third
rightwards double arrow fraction numerator 9 tan squared 10 degree minus 6 tan to the power of 4 10 degree plus tan to the power of 6 10 degree over denominator 1 minus 6 tan squared 10 degree plus 9 tan to the power of 4 10 degree end fraction equals 1 third
rightwards double arrow 27 tan squared 10 degree minus 18 tan to the power of 4 10 degree plus 3 tan to the power of 6 10 degree equals 1 minus 6 tan squared 10 degree plus 9 tan to the power of 4 10 degree
rightwards double arrow 3 tan to the power of 6 space 10 to the power of ring operator minus 27 tan to the power of 4 space 10 to the power of ring operator plus 33 tan squared space 10 to the power of ring operator equals 1

    Related Questions to study

    General
    Maths-

    sin space 20 to the power of ring operator open parentheses 4 plus sec space 20 to the power of ring operator close parentheses =

    space space space space space sin space 20 to the power of ring operator open parentheses 4 plus sec space 20 to the power of ring operator close parentheses
equals 4 sin space 20 degree plus fraction numerator sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 4 sin 20 degree. cos 20 degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 left parenthesis 2 sin space 20 degree. cos space 20 degree right parenthesis plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 sin space 40 degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 sin space left parenthesis 60 minus 20 right parenthesis degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 open parentheses sin space 60 degree cos space 20 degree minus cos space 60 degree sin space 20 degree close parentheses plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style cos space 20 degree minus begin display style 1 half end style sin space 20 degree close parentheses plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator square root of 3 cos space 20 degree minus sin space 20 degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator square root of 3 cos space 20 degree over denominator cos space 20 degree end fraction
equals square root of 3

    sin space 20 to the power of ring operator open parentheses 4 plus sec space 20 to the power of ring operator close parentheses =

    Maths-General
    space space space space space sin space 20 to the power of ring operator open parentheses 4 plus sec space 20 to the power of ring operator close parentheses
equals 4 sin space 20 degree plus fraction numerator sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 4 sin 20 degree. cos 20 degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 left parenthesis 2 sin space 20 degree. cos space 20 degree right parenthesis plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 sin space 40 degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 sin space left parenthesis 60 minus 20 right parenthesis degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 open parentheses sin space 60 degree cos space 20 degree minus cos space 60 degree sin space 20 degree close parentheses plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator 2 open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style cos space 20 degree minus begin display style 1 half end style sin space 20 degree close parentheses plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator square root of 3 cos space 20 degree minus sin space 20 degree plus sin space 20 degree over denominator cos space 20 degree end fraction
equals fraction numerator square root of 3 cos space 20 degree over denominator cos space 20 degree end fraction
equals square root of 3
    General
    Maths-

    In the figure P Q greater than P R comma QS and RS are the bisectors of stack ∣ Q with _ below and stack R with _ below respectively then


    PQ>PR
    =anglePRQ>anglePQR           [angle opposite to the longer side of triangle is greater]
    =1/2 anglePRQ> 1/2anglePQR
    =angleSRQ>angleSQR           [as QR is the bisector of anglePQR and RS is the bisector of anglePRQ]
    = SQ>SR             [as side opposite to greater angle is greatest]
    So, SQ>SR

    In the figure P Q greater than P R comma QS and RS are the bisectors of stack ∣ Q with _ below and stack R with _ below respectively then

    Maths-General

    PQ>PR
    =anglePRQ>anglePQR           [angle opposite to the longer side of triangle is greater]
    =1/2 anglePRQ> 1/2anglePQR
    =angleSRQ>angleSQR           [as QR is the bisector of anglePQR and RS is the bisector of anglePRQ]
    = SQ>SR             [as side opposite to greater angle is greatest]
    So, SQ>SR
    General
    Maths-

    If AB=AC and A C D equals 115 to the power of ring operator end exponent then stack B with _ below A C equals

    If AB=AC and A C D equals 115 to the power of ring operator end exponent then stack B with _ below A C equals

    Maths-General
    parallel
    General
    Maths-

    In the figure A equals 30 to the power of ring operator end exponent & E B equals 50 to the power of ring operator end exponent then ACD=

    In the figure A equals 30 to the power of ring operator end exponent & E B equals 50 to the power of ring operator end exponent then ACD=

    Maths-General
    General
    Maths-

    open parentheses sin to the power of 8 space 75 to the power of ring operator minus cos to the power of 8 space 75 to the power of ring operator close parentheses =

    space space space open parentheses sin to the power of 8 space 75 to the power of ring operator minus cos to the power of 8 space 75 to the power of ring operator close parentheses
equals sin to the power of 8 open parentheses 90 minus 15 close parentheses degree minus cos to the power of 8 left parenthesis 90 minus 15 right parenthesis degree
equals cos to the power of 8 15 degree minus sin to the power of 8 15 degree
equals open parentheses cos to the power of 4 15 degree plus sin to the power of 4 15 degree close parentheses open parentheses cos to the power of 4 15 degree minus sin to the power of 4 15 degree close parentheses
equals open parentheses open parentheses cos squared 15 degree plus sin squared 15 degree close parentheses squared minus 2 sin squared 15 degree. cos squared space 15 degree close parentheses open parentheses cos squared 15 degree plus sin squared 15 degree close parentheses open parentheses cos squared 15 degree minus sin squared 15 degree close parentheses
equals open parentheses 1 minus open parentheses 2 sin space 15 degree. cos 15 degree close parentheses squared over 2 close parentheses 1 cross times cos open parentheses 2 cross times 15 close parentheses degree
equals open parentheses 1 minus fraction numerator s i n squared 30 degree over denominator 2 end fraction close parentheses cos space 30 degree
equals open parentheses 1 minus 1 over 8 close parentheses fraction numerator square root of 3 over denominator 2 end fraction
equals 7 over 8. fraction numerator square root of 3 over denominator 2 end fraction
equals fraction numerator 7 square root of 3 over denominator 16 end fraction

    open parentheses sin to the power of 8 space 75 to the power of ring operator minus cos to the power of 8 space 75 to the power of ring operator close parentheses =

    Maths-General
    space space space open parentheses sin to the power of 8 space 75 to the power of ring operator minus cos to the power of 8 space 75 to the power of ring operator close parentheses
equals sin to the power of 8 open parentheses 90 minus 15 close parentheses degree minus cos to the power of 8 left parenthesis 90 minus 15 right parenthesis degree
equals cos to the power of 8 15 degree minus sin to the power of 8 15 degree
equals open parentheses cos to the power of 4 15 degree plus sin to the power of 4 15 degree close parentheses open parentheses cos to the power of 4 15 degree minus sin to the power of 4 15 degree close parentheses
equals open parentheses open parentheses cos squared 15 degree plus sin squared 15 degree close parentheses squared minus 2 sin squared 15 degree. cos squared space 15 degree close parentheses open parentheses cos squared 15 degree plus sin squared 15 degree close parentheses open parentheses cos squared 15 degree minus sin squared 15 degree close parentheses
equals open parentheses 1 minus open parentheses 2 sin space 15 degree. cos 15 degree close parentheses squared over 2 close parentheses 1 cross times cos open parentheses 2 cross times 15 close parentheses degree
equals open parentheses 1 minus fraction numerator s i n squared 30 degree over denominator 2 end fraction close parentheses cos space 30 degree
equals open parentheses 1 minus 1 over 8 close parentheses fraction numerator square root of 3 over denominator 2 end fraction
equals 7 over 8. fraction numerator square root of 3 over denominator 2 end fraction
equals fraction numerator 7 square root of 3 over denominator 16 end fraction
    General
    Maths-

    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 =

    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

    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 =

    Maths-General
    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
    parallel
    General
    Maths-

    If tan space left parenthesis pi cos space theta right parenthesis equals cot space left parenthesis pi sin space theta right parenthesis comma 0 less than theta less than fraction numerator 3 pi over denominator 4 end fraction then sin space open parentheses theta plus pi over 4 close parentheses equals

    space space space space tan left parenthesis pi space cos space ϑ right parenthesis equals c o t space left parenthesis pi space sin space ϑ right parenthesis
rightwards double arrow tan left parenthesis pi space cos space ϑ right parenthesis equals tan space open parentheses pi over 2 minus pi space sin space ϑ close parentheses
rightwards double arrow pi space cos space ϑ equals pi over 2 minus pi space sin space ϑ
rightwards double arrow cos space ϑ equals 1 half minus sin space ϑ
rightwards double arrow cos space ϑ plus sin space ϑ equals 1 half
M u l t i p l y i n g space b o t h space t h e space s i d e s space w i t h space fraction numerator 1 over denominator square root of 2 end fraction comma space w e space g e t colon
rightwards double arrow fraction numerator 1 over denominator square root of 2 end fraction cos space ϑ plus fraction numerator 1 over denominator square root of 2 end fraction sin space ϑ equals fraction numerator 1 over denominator 2 square root of 2 end fraction
rightwards double arrow sin space pi over 4 cos space ϑ plus cos space pi over 4 sin space ϑ equals fraction numerator 1 over denominator 2 square root of 2 end fraction
rightwards double arrow sin open parentheses ϑ plus pi over 4 close parentheses equals fraction numerator 1 over denominator 2 square root of 2 end fraction

    If tan space left parenthesis pi cos space theta right parenthesis equals cot space left parenthesis pi sin space theta right parenthesis comma 0 less than theta less than fraction numerator 3 pi over denominator 4 end fraction then sin space open parentheses theta plus pi over 4 close parentheses equals

    Maths-General
    space space space space tan left parenthesis pi space cos space ϑ right parenthesis equals c o t space left parenthesis pi space sin space ϑ right parenthesis
rightwards double arrow tan left parenthesis pi space cos space ϑ right parenthesis equals tan space open parentheses pi over 2 minus pi space sin space ϑ close parentheses
rightwards double arrow pi space cos space ϑ equals pi over 2 minus pi space sin space ϑ
rightwards double arrow cos space ϑ equals 1 half minus sin space ϑ
rightwards double arrow cos space ϑ plus sin space ϑ equals 1 half
M u l t i p l y i n g space b o t h space t h e space s i d e s space w i t h space fraction numerator 1 over denominator square root of 2 end fraction comma space w e space g e t colon
rightwards double arrow fraction numerator 1 over denominator square root of 2 end fraction cos space ϑ plus fraction numerator 1 over denominator square root of 2 end fraction sin space ϑ equals fraction numerator 1 over denominator 2 square root of 2 end fraction
rightwards double arrow sin space pi over 4 cos space ϑ plus cos space pi over 4 sin space ϑ equals fraction numerator 1 over denominator 2 square root of 2 end fraction
rightwards double arrow sin open parentheses ϑ plus pi over 4 close parentheses equals fraction numerator 1 over denominator 2 square root of 2 end fraction
    General
    Maths-

    The minimum value m space equals bold space 2 cos space theta plus fraction numerator 1 over denominator sin invisible function application theta end fraction plus square root of 2 tan space theta space text  in  end text open parentheses 0 comma pi over 2 close parentheses is

    m equals 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ
sin c e comma space ϑ element of open parentheses 0 comma pi over 2 close parentheses
S o comma space sin space ϑ greater than 0 comma space cos space ϑ greater than 0 comma tan space ϑ greater than 0
u sin g space A M thin space greater or equal than G M space p r o p e r t y.
rightwards double arrow fraction numerator 2 space cos space ϑ plus begin display style fraction numerator 1 over denominator sin space ϑ end fraction end style plus square root of 2 tan space ϑ over denominator 3 end fraction greater or equal than 3 open parentheses 2 square root of 2 close parentheses to the power of bevelled 1 third end exponent
rightwards double arrow 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ greater or equal than 3 open parentheses square root of 2 cubed close parentheses to the power of bevelled 1 third end exponent
rightwards double arrow 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ greater or equal than 3 square root of 2
S o comma space m i n i m u m space v a l u e space i s space 3 square root of 2.

    The minimum value m space equals bold space 2 cos space theta plus fraction numerator 1 over denominator sin invisible function application theta end fraction plus square root of 2 tan space theta space text  in  end text open parentheses 0 comma pi over 2 close parentheses is

    Maths-General
    m equals 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ
sin c e comma space ϑ element of open parentheses 0 comma pi over 2 close parentheses
S o comma space sin space ϑ greater than 0 comma space cos space ϑ greater than 0 comma tan space ϑ greater than 0
u sin g space A M thin space greater or equal than G M space p r o p e r t y.
rightwards double arrow fraction numerator 2 space cos space ϑ plus begin display style fraction numerator 1 over denominator sin space ϑ end fraction end style plus square root of 2 tan space ϑ over denominator 3 end fraction greater or equal than 3 open parentheses 2 square root of 2 close parentheses to the power of bevelled 1 third end exponent
rightwards double arrow 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ greater or equal than 3 open parentheses square root of 2 cubed close parentheses to the power of bevelled 1 third end exponent
rightwards double arrow 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ greater or equal than 3 square root of 2
S o comma space m i n i m u m space v a l u e space i s space 3 square root of 2.
    General
    Maths-

    In the adjoining figure, stack left floor a with _ below equals left floor b and left floor c equals left floor d comma then left floor b plus left floor c equals

    In the adjoining figure, stack left floor a with _ below equals left floor b and left floor c equals left floor d comma then left floor b plus left floor c equals

    Maths-General
    parallel
    General
    Maths-

    In the figure, a equals 4 x to the power of ring operator end exponent and b equals left parenthesis 2.5 x plus 24 right parenthesis to the power of ring operator end exponent comma then the value of x =

    In the figure, a equals 4 x to the power of ring operator end exponent and b equals left parenthesis 2.5 x plus 24 right parenthesis to the power of ring operator end exponent comma then the value of x =

    Maths-General
    General
    Maths-

    In the following figure, the value of x =

    In the following figure, the value of x =

    Maths-General
    General
    Maths-

    Using information given in the following figure, the value of x and y is

    Using information given in the following figure, the value of x and y is

    Maths-General
    parallel
    General
    Maths-

    In figure, the sides QP and RQ of triangle P Q R are produced to points S and T respectively. If stack S P R with _ below equals 135 to the power of ring operator end exponent and open floor P Q T equals 110 to the power of ring operator end exponent close then stack P R Q with _ below equals

    In figure, the sides QP and RQ of triangle P Q R are produced to points S and T respectively. If stack S P R with _ below equals 135 to the power of ring operator end exponent and open floor P Q T equals 110 to the power of ring operator end exponent close then stack P R Q with _ below equals

    Maths-General
    General
    Maths-

    In the given figure, if P Q R equals 70 to the power of ring operator end exponent then ∣ A B C equals


    PQ is parallel to AB and QR is the transversal
    Also, QR is parallel to BC and AB is the transversal.
    angle A B C equals angle Q O B equals angle P Q R = 70 degree
    As angles are alternate interior angles so the angles are equal.

    In the given figure, if P Q R equals 70 to the power of ring operator end exponent then ∣ A B C equals

    Maths-General

    PQ is parallel to AB and QR is the transversal
    Also, QR is parallel to BC and AB is the transversal.
    angle A B C equals angle Q O B equals angle P Q R = 70 degree
    As angles are alternate interior angles so the angles are equal.
    General
    Maths-

    From the given figure, the value of ‘a’ is


    As the AB is parallel to CD and there is transversal
    So, the vertical opposite angle of 50 degree as those are always equal.
    Now, we have the interior angles, and we know that their sum is 180 degree

    S o comma 50 degree plus a equals 180 degree
rightwards double arrow a equals 180 degree minus 50 degree
rightwards double arrow a equals 130 degree

    From the given figure, the value of ‘a’ is