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Please, check my work on Trig Identities

  1. Dec 16, 2011 #1
    Please, check my work.

    1. The problem statement, all variables and given/known data

    a) Show that sqrt{[1+tan^2x]/[1+cot^2x]}=tanx

    b) Show that [cosx+sinx]/[cosx-sinx]=1+[2tanx]/[1- tanx]

    c)Show that cotxcosx+tanxsinx=(cosecx+ secx)(1-sinxcosx)

    d) Show that cosec^2x-cosecx=cot^2x/[1+sinx]

    e) Show that sin^3x-cos^3x= (sinx-cosx)(1+sinxcosx)

    f) Show that [cosx-1]/[secx+tanx]+[cosx+1]/[secx-tanx]=2(1+tanx)

    2. Relevant equations
    3. The attempt at a solution

    a) sqrt{[1+tan^2x]/[1+cot^2x]}=tanx

    =sqrt{[1+tan^2x]/[1+cot^2x]}

    = sqrt{[sec^2x]/[cosec^2x]}

    = secx/cosecx

    = 1/cosx(sinx)

    = tanx

    b) [cosx+sinx]/[cosx-sinx]=1+[2tanx]/[1- tanx]

    =1+[2tanx]/[1- tanx]

    = [1-tanx+2tanx]/[1-tanx]

    = [1+tanx]/[1-tanx]

    = [1+ sinx/cosx]/[1-sinx/cosx]

    = [(cosx+sinx)/cosx]/[(cosx-sinx)/cosx]

    = [cosx+sinx]/[cosx-sinx)]

    c) cotxcosx+tanxsinx=(cosecx+ secx)(1-sinxcosx)

    =(cosecx+secx)(1-sinxcosx)

    =(cosecx+secx)(sin^2x-sinxcosx+cos^2x)

    = sinx+ [sin^2x/cosx]-cosx-sinx+[cos^2x/sinx]+cosx

    = [sin^2x/cosx]+[cos^2x/sinx]

    = [sinx/cosx](sinx)+[cosx/sinx](cosx)

    = tanxsinx+cotxcosx

    d) cosec^2x-cosecx=cot^2x/[1+sinx]

    =cot^2x/[1+sinx]

    = cot^2x/[1+sinx]*[(1-sinx)/(1-sinx)]

    = [cot^2x-cot^2x(sinx)]/(1-sin^2x)

    = [cot^2x-cot^2x(sinx)]/(cos^2x)

    = [cot^2x]/[cos^2x]- [cot^2xsinx]/[cos^2x]

    = [cos^2x]/[sin^2x](1/[cos^2x])- [cos^2x]/[sin^2x](1/[cos^2x])(sinx)

    = 1/(sin^2x)-sinx/(sin^2x)

    = cosec^2x-cosecx

    e) sin^3x-cos^3x= (sinx-cosx)(1+sinxcosx)

    = (sinx-cosx)(1+sinxcosx)

    =( sinx-cosx)(sin^2x+sinxcosx+cos^2x)

    = sin^3x-cos^3x

    f) [cosx-1]/[secx+tanx]+[cosx+1]/[secx-tanx]=2(1+tanx)

    = [cosx-1]/[secx+tanx]+[cosx+1]/[secx-tanx]

    = [cosx-1]/[secx+tanx]+{[cosx+1]/[secx-tanx]*[(secx+tanx)/secx+tanx]}

    = {(cosx-1)(secx-tanx)+(cosx+1)(secx+tanx)}/[sex^2x-tan^2x]

    = 1-sinx-(1/cosx)+(sinx/cosx)+1+sinx+(1/cosx)+sinx/cosx

    = 1+(sinx/cosx)+1+sinx/cosx

    = 2+2(sinx/cosx)

    = 2(1+tanx)

    Thank You very much.
     
  2. jcsd
  3. Dec 16, 2011 #2

    eumyang

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    Homework Helper

    The work was a little difficult for me to read, even with the grouping symbols. But looks like (a) through (e) are correct. I didn't get a chance to look at (f).
     
  4. Dec 16, 2011 #3

    SammyS

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    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    (f) looks fine !
     
  5. Dec 16, 2011 #4
    Thank you very much for checking my work, people. Much appreciated.
     
  6. Dec 25, 2011 #5
    You shouldn't really need your work checked. As long as you're confident you didn't break any rules of math and arrive at the final answer it is most likely correct.
     
  7. Dec 26, 2011 #6

    Mark44

    Staff: Mentor

    I think that was the point of the OP's post - he wasn't confident about his or her work. It takes quite a bit of practice to reach that point of confidence.
     
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