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Prove that: (cos2theta)/(1+sin2theta)=ta...

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

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`Cos 2 theta /(1+{Sin} 2 theta)`

`=(cos ^2 theta-sin ^2 theta) /(1+2 sin theta cos theta)`

`=(cos ^2 theta-sin ^2 theta) /(sin ^2 theta+cos ^2 theta+2 sin theta cos theta)`

`=(cos theta+sin theta)(cos theta-sin theta) /(sin theta+cos theta)^2`

`=(cos theta-sin theta) /(sin theta+cos theta)`

`=(1-tan theta) /(1+tan theta)`

`=(tan pi / 4-tan theta) /(1+tan pi / 4tan theta)` `as tan pi / 4=1`

`=Tan(pi / 4-theta)`

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RD SHARMA-TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES-Solved Examples And Exercises
  1. Prove that: (sin2theta)/(1+cos2theta+sintheta)=t a ntheta

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  2. Prove that sin(2theta)/(1-cos2theta)=cot theta

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

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  4. Prove that: (1+sin2theta+cos2theta)/(1+sin2theta-cos2theta)=cot theta

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

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

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  7. Prove that: sin10^0sin30^0sin50^0sin70^0=1/(16)dot

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  9. Prove that: (sin5x-2sin3x+sin x)/(cos5x-cos x)=tanx

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  10. (sinx-sin3x)/(sin^(2)x-cos^(2)x) = 2sinx

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  11. Express each of the following as the sum or difference of sines and ...

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  12. Show that sqrt(3)\ cos e c\ 20^0-sec20^0=4.

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  13. Prove that: tan4theta=(4t a ntheta(1-tan^2theta))/(1-6tan^2theta+tan^4...

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  14. If tanx=3/4,pi<x<(3pi)/2 , fid the values of sinx/2,cosx/2a n dtanx//2...

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  15. Find : cos 20^0cos 40^0cos 60^0cos 80^0

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

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

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

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

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

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