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Prove : (1+cosA - sinA)/(1+cosA + sinA) = secA - tanA

  1. Aug 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that
    (1+cosA - sinA)/(1+cosA + sinA) = secA - tanA



    2. Relevant equations
    sin^2A + cos^2A = 1
    tanA = sinA/cosA
    cotA = cosA/sinA
    1 + cot^2A = cosec^2A
    tan^2A + 1 = sec^2A
    cosecA = 1/sinA
    secA = 1/cosA
    cotA = 1/tanA
    (Only use the above identities to prove the question)

    3. The attempt at a solution
    I'm stumped at this question. I have attempted various methods using the formulas that I know(stated above)and also trying to work on both sides but to no avail. I understand that by cross multiplying we can easily prove it but the correct way seems to just be by making either the LHS or RHS equal to the other,respectively. Can anyone help?
     
  2. jcsd
  3. Aug 23, 2010 #2

    hunt_mat

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    Multiply the LHS by:
    [tex]
    \frac{1+\cos A-\sin A}{1+\cos A-\sin A}
    [/tex]
    Expand.
     
  4. Aug 23, 2010 #3
    Don't i need to account for the RHS also? Or are we rationalizing like we do for surds?
     
  5. Aug 23, 2010 #4

    hunt_mat

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    You're multiplying by 1, so you only need to do this for the LHS, expand ans you'll see that things cancel and you end up with the RHS
     
  6. Aug 23, 2010 #5
    I think i went wrong?

    I finalized to ,
    2+2cosA - 2sinA - 2sinAcosA
    ----------------------------
    1 + 2cosA + cos^2A - sin^2A

    Sorry if this is hard to read,i don't know how to use latex. :/
     
  7. Aug 23, 2010 #6

    hunt_mat

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    You're perfectly correct, you write 1=sin^{2}A+\cos^{2}A in the deominator, does the numorator factor (hint, it does).

    Mat
     
  8. Aug 23, 2010 #7
    Do you group the sin and the cos together before factoring? If so,where do we put the troublesome sinAcosA?

    i'm really bad at this. I only managed to factor the denominator to cosA(2+2cosA)
     
  9. Aug 23, 2010 #8

    hunt_mat

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    You're halfway there! Look for the factor (2-2cosA) in the numorator, and then they should cancel.

    Mat
     
  10. Aug 23, 2010 #9
    Okay wait i cheated a little by looking at my RHS that i have converted into a fraction and i got it. Thanks alot! the numerator factors into ( 1-sinA) ( 2+2cosA) am i right? :)
     
  11. Aug 23, 2010 #10

    hunt_mat

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    Well done. You've done it.
     
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