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Prove that: cos^2A+cos^2(A+(2pi)/3)+cos...

Prove that: `cos^2A+cos^2(A+(2pi)/3)+cos^2(A-(2pi)/3)=3/2`

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To prove that \[ \cos^2 A + \cos^2\left(A + \frac{2\pi}{3}\right) + \cos^2\left(A - \frac{2\pi}{3}\right) = \frac{3}{2} \] we will start from the left-hand side and simplify it step by step. ...
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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.

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RD SHARMA-TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES-Solved Examples And Exercises
  1. If 5 sinalpha=3"sin"(alpha+2beta)!=0, then tan(alpha+beta) is equal ...

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  2. If tantheta=a/b ,then bcos2theta+asin2theta is equal to (a) a (b) b ...

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  3. Prove that: cos^2A+cos^2(A+(2pi)/3)+cos^2(A-(2pi)/3)=3/2

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  4. If 2tanalpha=3t a nbeta,t h e ntan(alpha-beta)= (a)(sin2beta)/(5-cos2...

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

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  6. What is the value of cos36^@ .

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

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  8. Prove that: cos18^0=(sqrt(10+2sqrt(5)))/4 .

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  9. If t a ntheta/2=sqrt((1-e)/(1+e))tan(alpha/2), then cosalpha= (a) 1-...

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  10. If cos2x+2cosx=1t h e n ,(2-cos^2x)sin^2x is equal to

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  11. If alphaa n dbeta are acute angles satisfying cos2alpha=(3cos2beta-1)/...

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  12. Prove that: cos^2A+cos^2(A+pi/3)+cos^2(A-pi/3)=3/2

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  13. Prove that: cos^(2)48^(@)-sin^(2)12^(@)=(sqrt(5)+1)/(8)

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  14. The value of (sin5alpha-sin3alpha)/(cos5alpha+2cos4alpha+cos3alpha) is

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  15. If costheta=1/2(a+1/a),cos3theta=lambda(a^3+1/(a^3)),t h e nlambda= ...

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  16. Prove that: t a nA+tan(60^0+A)-t a n(60^0-A)=3t a n3A

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  17. If t a n^2theta=2tan^2phi+1 , prove that cos2theta+sin^2phi=0.

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  18. If theta=pi/(2^n+1) , prove that: 2^ncosthetacos2thetacos2^2 cos2^(n-1...

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

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  20. Prove that: cos^2 48^0-sin^2 12^0=(sqrt(5)+1)/8

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