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sin ^(4) "" (pi)/(8) + sin ^(4) "" (3pi)...

`sin ^(4) "" (pi)/(8) + sin ^(4) "" (3pi)/(8) + sin ^(4) "" (5pi)/(8) + sin ^(4)"" (7pi)/(8) = (3)/(2).`

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Prove that: cos ^(4) ""(pi)/(8) + cos ^(4) ""(3pi)/(8) + sin ^(4) "" (5pi)/(8) + sin ^(4)""(7pi)/(8) = 3/2.

Prove that cos ^(4) ""(pi)/(8) + cos ^(4) ""(3pi)/(8) + cos ^(4) ""(5pi)/(8) + cos ^(4) "" (7pi)/(8) = (3)/(2)

sin "" (pi)/(5) sin "" (2pi)/(5) sin ""(4pi)/(5) sin""(3pi)/(5) = (5)/(16).

Evaluate : cos ""(pi)/(8) sin ""(pi)/(8)

Prove that sin ""(pi)/(9) sin ""(2pi)/(9) sin ""( 3pi)/(9) sin ""(4pi)/(9) = (3)/(16).

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

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) is

Evaluate : sin^(2) (pi/8 +x/2) - sin^(2) (pi/8 - x/2)

Prove that: sin^2(pi/8)+sin^2((3pi)/8)+sin^2((5pi)/8)+sin^2((7pi)/8)=2

ICSE-COMPOUND AND MULTIPLE ANGLES -EXERCISE 5 (D)
  1. Prove that (cos A - cos B) ^(2) + (sin A - sin B ) ^(2) = 4 sin ^(2) (...

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  2. Prove that: cos ^(4) ""(pi)/(8) + cos ^(4) ""(3pi)/(8) + sin ^(4) "" (...

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  3. sin ^(4) "" (pi)/(8) + sin ^(4) "" (3pi)/(8) + sin ^(4) "" (5pi)/(8) +...

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  4. cos alpha cos(60^(@) - alpha ) cos (60^(@) + alpha ) = 1/4 cos 3 alpha...

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  5. (cos ^(3) alpha - cos 3 alpha )/( cos alpha ) + (sin ^(3)alpha + sin 3...

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  6. cos ^(2) 2 theta - sin ^(2) theta = cos theta . cos 3 theta.

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  7. 1 + (cos 2 theta + cos 6 theta)/(cos 4 theta) = (sin 3 theta)/(sin t...

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  8. Prove that cos ^(3) (x-(2pi)/(3)) + cos ^(3) x + cos ^(3) (x + (2pi)/(...

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  9. (1)/(sin 10^(@)) - (sqrt3)/(cos 10 ^(@)) = 4.

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  10. tan 70^(@) -tan 20^(@) - 2 tan 40^(@) = 4 tan 10 ^(@).

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  11. tan theta + 2 tan 2 theta + 4 tan 4 theta + 8 cot 8 theta = cot theta.

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  12. 4 (cos ^(3) 20^(@) + cos ^(3) 40^(@)) = 3 ( cos 20 ^(@) + cos 40^(@))

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  13. Prove that 4(cos ^(3) 10^(@) + sin ^(3) 20^(@)) = 3 ( cos 10^(@) + sin...

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  14. Prove that cos ^(3) x sin ^(2) x = (1)/(16) (2cos x - cos 3x - cos 5x)...

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  15. cot A + cot (60^(@) + A) + cot (120^(@) +A) = 3 cot 3 A.

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  16. Prove that : cos 36 ^(@) - sin 18^(@) = (1)/(2).

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  17. cos ^(2) 36^(@) + sin ^(2) 18^(@) = (3)/(4).

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  18. 3 cos 72 ^(@) -4 sin ^(3) 18^(@) = cos 36 ^(@).

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  19. cos ^(2) 48^(@) - sin ^(2) 12 ^(@) = (sqrt5+1)/(8).

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  20. Prove that sin "" (1)/(10) pi cos ""(1)/(5) pi = (1)/(4).

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