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Diana0920

  • 2 years ago

cot(theta)-tan(theta) over sin(theta) + cos(theta) equals csc(theta) - sec(theta) how do I proof this?

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  1. Mertsj
    • 2 years ago
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    Let's use x instead of theta

  2. Diana0920
    • 2 years ago
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    okay

  3. Mertsj
    • 2 years ago
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    \[\frac{\cot x-\tan x}{\sin x+\cos x}=\csc x-\sec x\]

  4. Mertsj
    • 2 years ago
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    Do I have the problem stated correctly?

  5. Diana0920
    • 2 years ago
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    yes :)

  6. Diana0920
    • 2 years ago
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    would I have to replace cot(x) with cos(x)/sin(x)

  7. Diana0920
    • 2 years ago
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    and tan(x) with sin(x)/cos(x)

  8. Mertsj
    • 2 years ago
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    \[(\frac{\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}}{\sin x+\cos x})\times \frac{\sin x \cos x}{\sin x \cos x}=\frac{\cos ^2x-\sin ^2x}{\sin x \cos x(\sin x+\cos x)}\]

  9. Mertsj
    • 2 years ago
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    Can you get it from there?

  10. Diana0920
    • 2 years ago
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    sadly no? can you explain how I can simplify it more please?

  11. Mertsj
    • 2 years ago
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    Factor the numerator and then the cosx + sinx will cancel.

  12. alejandrop95
    • 2 years ago
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    I'm still lost :(

  13. Mertsj
    • 2 years ago
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    @alejandrop95 This is not your question

  14. Diana0920
    • 2 years ago
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    how would I factor it though? I'm sorry I am new to trig.

  15. Mertsj
    • 2 years ago
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    Can you factor this: \[a^2-b^2\]

  16. Diana0920
    • 2 years ago
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    I don't think you can because it doesn't have the same coefficient

  17. Mertsj
    • 2 years ago
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    \[a^2−b^2=(a-b)(a+b)\]

  18. Mertsj
    • 2 years ago
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    \[\cos ^2x-\sin ^2x=(\cos x-\sin x)(\cos x+\sin x)\]

  19. Diana0920
    • 2 years ago
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    oh okay that makes sense. so would it be cos-sin(cos+sin)

  20. Mertsj
    • 2 years ago
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    yes

  21. Diana0920
    • 2 years ago
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    then that would cancel with the denominator correct?

  22. Mertsj
    • 2 years ago
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    Yes. sinx + cosx will cancel

  23. Diana0920
    • 2 years ago
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    which will leave 1/sin(x)+cos(x)

  24. Mertsj
    • 2 years ago
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    no

  25. Diana0920
    • 2 years ago
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    using the reciprocal identities this will equal csc(x)-sec(x) ?

  26. Mertsj
    • 2 years ago
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    |dw:1382579245755:dw|

  27. Diana0920
    • 2 years ago
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    cos(x)-sin(x)/sin(x)+cos(x0

  28. Diana0920
    • 2 years ago
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    this is where I would use the reciprocal identities?

  29. Mertsj
    • 2 years ago
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    |dw:1382579370642:dw|

  30. Diana0920
    • 2 years ago
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    the cos(x) cancel and the sin(x) cancels?

  31. Mertsj
    • 2 years ago
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    |dw:1382579509266:dw|

  32. Diana0920
    • 2 years ago
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    yay!!!! the cancel and its one,which now you are left with the csc(x) and sec(x) reciprocal identities!! Thank you!! the key idea here that I need to work on was the factoring

  33. Diana0920
    • 2 years ago
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    thank you !:)

  34. alejandrop95
    • 2 years ago
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    thanks ! :)

  35. Mertsj
    • 2 years ago
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    yw

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