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prove sin(5theta) = 16sin^5(theta)-20si...

prove ` sin(5theta) = 16sin^5(theta)-20sin^3(theta)+5sin(theta)`

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Given,
`sin5theta=sin(3theta+2theta)
=sin3thetacos2theta+cos3thetasin2theta`
`=(3sintheta−4sin^3theta)(1–2sin^2theta)+cos(2theta+theta)sin2theta`
`=3sintheta−10sin^3theta+8sin5theta+[cos2thetacostheta−sin2thetasintheta]sin2theta`
`=3sintheta−10sin^3theta+8sin5theta+[(1–2sin^2theta)costheta−2sin^2thetacostheta]2sinAcosA`
`=3sinA−10sin³A+8sin5A+[cosA−4sin²AcosA]2sinAcosA`
`=3sinA−10sin³A+8sin5A+2sinAcos²A−8sin³Acos²A`
...
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  • TRIGONOMETRIC RATIOS OF COMPOUND ANGLES

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RD SHARMA-TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES-Solved Examples And Exercises
  1. Prove that: sin A sin(60-A)sin(60+A)=1/4sin3Adot

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  2. Prove that: "sin"20^0"sin"40^0"sin"60^0"sin"80^0=3/(16)

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  3. prove sin(5theta) = 16sin^5(theta)-20sin^3(theta)+5sin(theta)

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  4. Prove the following identity: 4(cos^3 10^0+sin^3\ \ 20^0)=\ 3\ (cos 10...

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  5. Prove the following identity: cos^3thetas in3theta+sin^3thetacos3theta...

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  6. Prove the following identity: tan θ tan(60^@ +θ)tan(60^@ −θ)=]ta...

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  7. Prove that: t a nA+tan(60^0+A)-t a n(60^0-A)=3t a n3ALet's use the sum...

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  8. Prove that: cotA+"cot"(60^0+A)-cot(60^0-A)=3cot3A

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  9. Prove the following identity: cot A+cot(60^0+A)+cot(120^0+A)=3cot3A

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  10. Prove that: sin^3A+sin^3((2pi)/3+A)+sin^3((4pi)/3+A)=-3/4sin3Adot

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  11. Maximum value of |costhetacos(60^0-theta)cos(60^0+theta) for all valu...

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  12. Prove that: sin(pi/5)sin(2pi/5)sin(3pi/5)sin(4pi/5)=5/16

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  13. Prove that cos(2pi)/(15)cos(4pi)/(15)cos(8pi)/(15)cos(14pi)/(15)=1/(16...

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  14. Prove that: sin 12^0sin 48^0sin 54^0=1/8dot

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  15. The value oif the expressin (1+cospi/10)(1+cos(3pi)/10)(1+cos(7pi)/10)...

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  16. Prove that ; 4 sin 27^@=(5+sqrt(5))-sqrt((3-sqrt(5))) we have

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  17. Prove that : sin^2(72^@) - sin^2 (60^@) = (sqrt5 - 1)/8

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  18. Prove that: sin^2 24^0-sin^2 6^0=(sqrt(5)-1)/8

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  19. Prove that: sin^2 42^0-cos^2 78^0=(sqrt(5)+1)/8

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  20. Prove that: cos 78^0cos 42^0cos 36^0=1/8

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