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Prove that : cos Acos(60-A)cos(60+A)=1/4...

Prove that `: cos Acos(60-A)cos(60+A)=1/4cos3A`

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To prove that \( \cos A \cos(60^\circ - A) \cos(60^\circ + A) = \frac{1}{4} \cos 3A \), we will follow these steps: ### Step 1: Use the product-to-sum identities We start with the left-hand side of the equation: \[ \cos A \cos(60^\circ - A) \cos(60^\circ + A) \] Using the product-to-sum identity for cosine, we know that: ...
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Prove that: cos A cos(60^(@)-A)cos(60^(@)+A)=(1)/(4)cos3A

Prove that: cosA. Cos(60^(@)+A)cos(60^(2)-A)=1/4 cos3A

(ii) Prove cos x cos(60-x)cos(60+x)=(1)/(4)cos3x

Prove that: (i) cos A cos(60^(@)-A)cos (60^(@)+A)=(1)/(4)cos 3A (ii) "tan" A+"tan"(60^(@)+A)-"tan"(60^(@)-A)=3 tan 3A .

Prove that: cos A+cos3A+cos5A+cos7A=4cos A cos2A cos4A

Prove that: |cos theta cos(60^(@)-theta)cos(60^(@)+theta)<=(1)/(4) for all values of theta

Prove that: (cos(60^(@)-A)+sin(30^(@)-A))/(cos(30^(@)-A)-sin(60^(@)-A))= cotA

Prove that: (cos(60^(@)-A)+sin(30^(@)-A))/(cos(30^(@)-A)-sin(60^(@)-A))=cotA

Prove that: cos3A+cos5A+cos7A+cos15A=4cos4A cos5A cos6A

Prove that : cos^(2)A+cos^(2)(A+pi/3)+cos^(2)(A-pi/3) =1/2[2cos^(2)A+2cos^(2)(A+60^(@))+2cos^(2)(A-60^(@))] =1/2[1+cos2A+1+cos(2A+120^(@))+1+cos(2A-120^(@))] 1/2[3+cos2A+cos(2A+120^(@))+cos(2A-120^(@))] 1/2[3+cos2A+2cos(2A+120^(@)+2A-120^(@))/(2).cos(2A+120^(@)-2A+120^(@))/(2)] =1/2[3+cos2A+2cos2Acos120^(@)] =1/2[3cos2A+2cos2Acos(90^(@)+30^(@))] =1/2[3+cos2A-2.cos2A.(1/2)] =1/2(3+cos2A-cos2A) =3/2 = RHS Hence Proved.

RD SHARMA-TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES-Solved Examples And Exercises
  1. Prove that: 108sinpi/(18)-144sin^3pi/(18)=18

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  2. Prove that cos6x=32cos^6x-48cos^4x+18cos^2x-1

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  3. Prove that : cos Acos(60-A)cos(60+A)=1/4cos3A

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  4. Prove that: sin A sin(60-A)sin(60+A)=1/4sin3Adot

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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