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kengeta Group Title

how do i prove : tan^2 θ cos^2 θ + cos^2 θ = 1

  • 4 months ago
  • 4 months ago

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  1. myininaya Group Title
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    well my first attempt to prove this is true is to take the left hand side and try to show the right hand side the left hand side has both terms with the factor cos^2(theta) start by factoring cos^2(theta) out from both terms this will get us closer to writting as 1 term just as the right hand side is (one term)

    • 4 months ago
  2. myininaya Group Title
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    let me know if you still don't know where to go after that

    • 4 months ago
  3. kengeta Group Title
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    I'm pretty confused

    • 4 months ago
  4. myininaya Group Title
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    on how to factor?

    • 4 months ago
  5. myininaya Group Title
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    do you know hot to factor the expression ax+x?

    • 4 months ago
  6. myininaya Group Title
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    how* (not hot)

    • 4 months ago
  7. kengeta Group Title
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    no

    • 4 months ago
  8. myininaya Group Title
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    I will show you how to factor ax+x then you should be able to factor tan^2(theta)*cos^2(theta)+cos^2(theta) so ax+x I see there is a x in both terms |dw:1407522006591:dw| so i will factor that x out like so \[ax+1x=x(a+1)\]

    • 4 months ago
  9. myininaya Group Title
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    you have \[\tan^2(\theta) \cdot \cos^2(\theta)+\cos^2(\theta) \cdot 1 \] \[\text{ do you see that there is a } cos^2(\theta) \text{ in both terms ?}\]

    • 4 months ago
  10. kengeta Group Title
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    yes

    • 4 months ago
  11. myininaya Group Title
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    can you factor that out using what you know about multiplication and division like i did above?

    • 4 months ago
  12. myininaya Group Title
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    also this way i'm asking you to approach it is not the only approach to take

    • 4 months ago
  13. myininaya Group Title
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    ok if you are having trouble factoring the cos^2(theta) out how about we try this another way do you recall tan is sin/cos?

    • 4 months ago
  14. kengeta Group Title
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    yes

    • 4 months ago
  15. myininaya Group Title
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    \[\frac{\sin^2(\theta)}{\cos^2(\theta)} \cos^2(\theta)+\cos^2(\theta)\] in the first term there, do you see anything that cancels?

    • 4 months ago
  16. kengeta Group Title
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    cos^2?

    • 4 months ago
  17. myininaya Group Title
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    right so what does that leave us with?

    • 4 months ago
  18. myininaya Group Title
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    \[\frac{\sin^2(\theta)}{\cancel{\cos^2(\theta)}}\cancel{\cos^2(\theta)}+\cos^2(\theta)\] what does this give us?

    • 4 months ago
  19. kengeta Group Title
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    sin^2(theta)+cos^2(theta)

    • 4 months ago
  20. myininaya Group Title
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    And that equals?

    • 4 months ago
  21. myininaya Group Title
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    you need to recall some trig identities for these trigonometric proofs

    • 4 months ago
  22. myininaya Group Title
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    This one of the most basic ones It actually has a name

    • 4 months ago
  23. kengeta Group Title
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    okay si... 1?

    • 4 months ago
  24. myininaya Group Title
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    I will give you a hint: Pythagorean Identity

    • 4 months ago
  25. myininaya Group Title
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    yep

    • 4 months ago
  26. kengeta Group Title
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    *so

    • 4 months ago
  27. kengeta Group Title
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    okay i get it now thanks soooo much

    • 4 months ago
  28. myininaya Group Title
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    Way 1: \[\tan^2(\theta) \cdot \cos^2(\theta)+\cos^2(\theta) \cdot 1= \\ \cos^2(\theta)(\tan^2(\theta)+1)= \\ \cos^2(\theta) \cdot \sec^2(\theta)= \\ 1\] Way 2: \[\tan^2(\theta) \cos^2(\theta)+\cos^2(\theta)= \\ \frac{\sin^2(\theta)}{\cos^2(\theta)} \cos^2(\theta)+\cos^2(\theta) = \\ \sin^2(\theta) +\cos^2(\theta)= \\ 1\] These are the two ways that I can think of... This does not mean they are the only ways.

    • 4 months ago
  29. myininaya Group Title
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    You don't have to do both ways. Just one way.

    • 4 months ago
  30. myininaya Group Title
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    I used a Pythagorean identity in both ways.

    • 4 months ago
  31. myininaya Group Title
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    way 1 was the way I was trying to get you to go about it the first time around

    • 4 months ago
  32. myininaya Group Title
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    I don't think one way is more harder than the other But you will have to review some algebra especially if you don't remember how to factor. Factoring will come up again.

    • 4 months ago
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