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Show that: sin 50^0cos85^0=(1-sqrt(2)sin...

Show that: `sin 50^0cos85^0=(1-sqrt(2)sin35^0)/(2sqrt(2))`

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`sin 50^{circ} cos 85^{circ}`
` frac{1}{2}(2 {Sin} 50^{circ} {Cos} 85^{circ}) `
` =frac{1}{2}[sin (50^{circ}+85^{circ})+sin (50^{circ}-85^{circ})] `
` because 2 {Sin} A {Cos} B={Sin}(A+B)+{Sin}(A- B) `
` =frac{1}{2}[frac{1}{sqrt{2}}-sin 35^{circ}] `
` =frac{1-sqrt{2} sin 35^{circ}}{2 sqrt{2}}`
`
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RD SHARMA-TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES-Solved Examples And Exercises
  1. Find : cos 20^0cos 40^0cos 60^0cos 80^0

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  2. Prove that : cos(pi/7)cos((2pi)/7) cos((3pi)/7)=1/8

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  3. Show that: sin 50^0cos85^0=(1-sqrt(2)sin35^0)/(2sqrt(2))

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

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  5. sin(pi/18)*sin(5pi/18)*sin(7pi/18)

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  6. cos(pi/15)cos((2pi)/15)cos((3pi)/15)cos((4pi)/15)cos((5pi)/15)cos((6pi...

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  7. If sinalpha+sinbeta=a and cosalpha+cosbeta=b prove that: cos(alpha-bet...

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  8. sinalpha+sinbeta=a ,cosalpha+cosbeta=b=>sin(alpha+beta)

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  9. Prove that sqrt((1-cos2theta)/(1+cos2theta))=tantheta where tantheta>0

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  10. Prove that: (sin2theta)/(1-cos2theta)=cottheta

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  11. Prove that: (sin2theta)/(1+cos2theta)=t a ntheta

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  12. Prove that: \ sqrt(2+sqrt(2+2cos4theta))=2costheta

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  13. (1+sin2theta-cos2theta)/(1+sin2theta+cos2theta)=tantheta

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  14. (sintheta +sin2theta)/(1+costheta+cos2theta)=tantheta

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  15. Prove that: (cos2theta)/(1+sin2theta)=tan(pi/4-theta)

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  16. Prove that: (costheta)/(1+sintheta)=tan(pi/4-theta/2)

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

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  18. Prove that: (cosalpha+cosbeta)^2+(s inalpha+s inbeta)^2=4cos^2\ \ ((al...

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  19. Prove that: 1+cos^2 2x=2(cos^4x+sin^4x)

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  20. cos^3 2theta+3cos2theta=4(cos^6theta-sin^6 theta)

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