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sin^(-1)[tan""(pi)/(4)]-sin^(-1)[sqrt((3...

`sin^(-1)[tan""(pi)/(4)]-sin^(-1)[sqrt((3)/(x))]=(pi)/(6).` Then x is a root of the equation

A

`x^2 -x-6=0`

B

`x^2 -x-12=0`

C

` x^2 +x -12=0`

D

`x^2 +x+6=0`

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The correct Answer is:
B
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To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))xx (cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is

To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))/(cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is

If sin^(-1)x+sin^(-1)y=(2pi)/(3)",then"cos^(-1)x+cos^(-1)y is equal to

FULL MARKS-INVERSE TRIGONOMETRIC FUNCTIONS-EXERCISE -4.6
  1. The value of sin^(-1) (cos x ) , 0 le x le pi is :

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  2. If sin^(-1)x+sin^(-1)y=(2pi)/(3)",then"cos^(-1)x+cos^(-1)y is equal to

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  3. sin^(-1)""(3)/(5)-cos^(-1)""(12)/(13)+sec^(-1)""(5)/(3)-cosec^(-1)(13)...

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  4. If sin^(-1)x=2sin^(-1)alpha has a solution, then

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  5. sin^(-1)(cosx)=(pi)/(2)-x is valid for

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  6. If sin^(-1)x+sin^(-1)y+sin^(-1)z=(3pi)/(2) , the value of x^(2017)+y^(...

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  7. If cot^(-1) x = (2pi)/7 for some x in R the value of tan^(-1) x is...

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  8. The domain of the function defined by f(x)=sin^(-1)sqrt(x-1) is

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  9. If x=(1)/(5),"the value of"cos(cos^(-1)x+2sin^(-1)x) is

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  10. tan^(-1)((1)/(4))+tan^(-1)((2)/(9)) is equal to

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  11. If the function f(x)=sin^(-1)(x^(2)-3)" then "x belongs to

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  12. If cot^(-1)2andcot^(-1)3 are two anges of a triangle, then the third a...

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  13. sin^(-1)[tan""(pi)/(4)]-sin^(-1)[sqrt((3)/(x))]=(pi)/(6). Then x is a ...

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  14. sin^(-1)(2cos^(2)x-1)+cos^(-1)(1-2sin^(2)x)=

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  15. If cot^(-1)(sqrt(sinalpha))+tan^(-1)(sqrt(sinalpha))=u,"then"cos2u"is ...

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  16. sin^(-1) ((2x)/(1 + x^(2)))

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  17. The equation tan^(-1)x-cot^(-1)x=tan^(-1)((1)/(sqrt(3))) has

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  18. If cos^(-1)x + cot^(-1) (1/2) = pi/2 then x is equal to

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  19. If sin^(-1)""(x)/(5)+cosec^(-1)""(5)/(4)=(pi)/(2), then the value of x...

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  20. sin(tan^(-1)x)absxlt1 is equal to

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