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sabika13

  • 2 years ago

how do prove this identity: sin^4x -cos^4x=1-2cos^2x

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  1. zordoloom
    • 2 years ago
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  2. sabika13
    • 2 years ago
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    RS: (sin^2x+cos^2x)-2cos^2x sin^2x-cos^2x Thats how much i can simplify right side..

  3. sabika13
    • 2 years ago
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    yah

  4. zordoloom
    • 2 years ago
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    alright.

  5. zordoloom
    • 2 years ago
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    So here is how I proved it:

  6. zordoloom
    • 2 years ago
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    So, change sinx^4 to (1-cosx^2)^2

  7. zordoloom
    • 2 years ago
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    You can expand that and you get 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2

  8. zordoloom
    • 2 years ago
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    Do you follow so far?

  9. zordoloom
    • 2 years ago
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    @sabika13 Should I continue?

  10. sabika13
    • 2 years ago
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    yes sorry

  11. zordoloom
    • 2 years ago
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    ok

  12. zordoloom
    • 2 years ago
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    When you simplify 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2, the left side becomes 1-2cosx^2=1-2cosx^2

  13. zordoloom
    • 2 years ago
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    That proves the identity.

  14. zordoloom
    • 2 years ago
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    Normally, when you prove an identity, you work on one side. In this problem I worked on the left side.

  15. sabika13
    • 2 years ago
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    wait how does (1-2cosx^2)^2=1-2cos^2+cosx^4?

  16. zordoloom
    • 2 years ago
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    To summarize it: You have sin(x)^4-cos(x)^4=1-2cos(x)^2 change sin(x)^4 --->(1-cos(x)^2)^2 You have: (1-cos(x)^2)^2-cos(x)^4=1-2cos(x)^2 Then expand (1-cos(x)^2)^2 on the left side. You get 1-2cos(x)^2+cos(x)^4-cos(x)^4=1-2cos(x)^2

  17. zordoloom
    • 2 years ago
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    Any better?

  18. zordoloom
    • 2 years ago
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    Then the: cos(x)^4-cos(x)^4 cancel out.

  19. zordoloom
    • 2 years ago
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    Hope that helped.

  20. sabika13
    • 2 years ago
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    im having problems expanding (1-cos^2x)^2 shouldnt it be: 1-cos^4x :S

  21. zordoloom
    • 2 years ago
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    So, (1-cos^2x)(1-cos^2x). Just foil out.

  22. sabika13
    • 2 years ago
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    ohhh okay, but why is it not: (1-cos^2x)(1+cos^2x)

  23. zordoloom
    • 2 years ago
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    It's squared. It's not a difference of squares.

  24. sabika13
    • 2 years ago
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    ohh i think i get it...

  25. zordoloom
    • 2 years ago
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    Any number, or equation squared is the same as it times by itself. Example (6)^2= (6)*(6) or (3x-2)^2= (3x-2)(3x-2). Do you get the point?

  26. zordoloom
    • 2 years ago
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    You don't change the signs inside the equation.

  27. sabika13
    • 2 years ago
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    yeah so whats difference of squares then (if you dont mind me asking)

  28. zordoloom
    • 2 years ago
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    example x^2-4. When you factor it, you get (x-2)(x+2). It's a^2-b^2

  29. sabika13
    • 2 years ago
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    ohhh thankkyou!!you helped me fix a very big misconception inmy head ;P

  30. zordoloom
    • 2 years ago
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    yep, no prob.

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