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Prove that sin (45^(@) + theta) - sin ...

Prove that
sin `(45^(@) + theta) - sin (45^(@) - theta) = sqrt(2) sin theta`

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The correct Answer is:
`sqrt(2) sin theta`
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Prove that sin (theta)/(2) sin"" (7 theta)/(2) + sin ""(3theta)/(2) sin ""(11 theta)/(2) = sin 2theta sin5 theta .

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

SURA PUBLICATION-TRIGONOMETRY-EXERCISE 3.4
  1. Find a quadratic equation whose roots are sin 15^(@) and cos 15^(@)

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  2. Expand cos ( A + B + C). Hence prove that cos A cos B cos C = sin A si...

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  3. Prove that sin (45^(@) + theta) - sin (45^(@) - theta) = sqrt(2) sin...

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  4. Prove that sin (30^(@) + theta) + cos (60^(@) + theta) = cos theta

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  5. If a cos (x + y) = b cos (x - y), show that (a + b)tan x = (a - b) cot...

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  6. Prove that sin 105^(@) + cos 105^(@) = cos 45^(@)

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  7. Prove that sin 75^(@) - sin 15^(@) = cos 105^(@) + cos 15^(@)

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  8. Show that tan 75^(@) + cot 75^(@) = 4

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  9. Prove that cos (A + B) cos C - cos (B + C) cos A = sin B sin (C - A)

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  10. Prove that sin (n + 1)theta sin (n - 1) theta + cos (n + 1)theta cos (...

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  11. If x cos theta = y cos (theta + (2pi)/(3))= z cos (theta + (4pi)/(3)) ...

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  12. Prove that sin ( A + B) sin (A - B) = sin^(2) A - sin^(2)B.

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  13. Prove that cos (A + B) cos (A - B) = cos^(2) A - sin^(2) B = cos^(2) ...

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  14. Prove that sin^(2)(A + B) - sin^(2)(A - B) = sin 2A sin 2B

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  15. Prove that cos 8theta cos 2theta = cos^(2) 5theta- sin^(2) 3 theta

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  16. Show that cos^(2) A + cos^(2)B - 2 cos A cos B cos (A + B) = sin^(2) (...

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  17. If cos (alpha - beta) + cos (beta - gamma) + cos (gamma - alpha) = (-3...

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  18. Show that tan(45^(@) + A) = (1 + tan A)/(1 - tan A)

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  19. Show that tan (45^(@) - A) = (1 - tanA)/(1 + tanA)

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  20. Prove that cot (A + B) = (cot A cot B - 1)/(cot A + cot B)

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