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Math Help - Trigonometry Identities?

  1. #1
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    Cool Trigonometry Identities?

    Prove each of the following identities algebraically:

    a) cos(x) + (cos(x)*tan(x)^2 = sec(x)

    b) cos(x)(sec(x)-csc(x))=1-cot(x)

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  2. #2
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    Quote Originally Posted by brewerangel View Post
    Prove each of the following identities algebraically:

    a) cos(x) + (cos(x)*tan(x)^2 = sec(x)

    b) cos(x)(sec(x)-csc(x))=1-cot(x)

    Hello,

    to a)
    Code:
                                         cos(x) * (sin(x))▓
    cos(x) + cos(x)*(tan(x))▓ = cos(x) + ------------------- =
                                            (cos(x))▓
    
             (sin(x))▓    cos(x)*cos(x) + (sin(x))▓
    cos(x) + --------- = ------------------------- = 
              cos(x)               cos(x)
    
         1
     ---------- = sec(x)
       cos(x)
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  3. #3
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    Quote Originally Posted by brewerangel View Post
    Prove each of the following identities algebraically:

    a) cos(x) + (cos(x)*tan(x)^2 = sec(x)
    cos(x) + cos(x) tan(x)^2 = cos(x) + cos(x) sin^2(x)/cos^2(x)

    ................. = cos(x) + sin^2(x)/cos(x) = [cos^2(x) + sin^2(x)]/cos(x)

    ................. = 1/cos(x) = sec(x)


    b) cos(x)(sec(x)-csc(x)) = 1-cot(x)
    cos(x)(sec(x)-csc(x)) = cos(x) [1/cos(x) - 1/sin(x)]

    .................... = cos(x) [ (sin(x) - cos(x))/(cos(x) sin(x) ]

    .................... = (sin(x) - cos(x))/ sin(x)

    .................... = 1 - cos(x)/sin(x) = 1- cot(x)

    RonL
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  4. #4
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    Quote Originally Posted by brewerangel View Post
    Prove each of the following identities algebraically:

    a) cos(x) + (cos(x)*tan(x)^2 = sec(x)

    b) cos(x)(sec(x)-csc(x))=1-cot(x)

    Hello,

    to b)
    Code:
                                        1         1
    cos(x)(sec(x) - csc(x)) = cos(x)*(------- - -------) =
                                       cos(x)   sin(x)
    
    cos(x)   cos(x)
    ------ - ------ = 1 - cot(x)
    cos(x)   sin(x)
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  5. #5
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    Hello, brewerangel!

    Another approach . . .


    a) .cos(x) + cos(x)Ětan▓(x) .= .sec(x)
    The left side is: .cos(x) + cos(x)Ětan▓(x)

    Factor: . . . . . . .cos(x)Ě[1 + tan▓(x)]

    Then we have: . cos(x)Ěsec▓(x) .= .cos(x)Ěsec(x)Ěsec(x)


    Since cos(x)Ěsec(x) = 1, we have: .sec(x)




    b) .cos(x)[sec(x)-csc(x)] .= .1 - cot(x)
    The left side is: .cos(x)Ěsec(x) - cos(x)Ěcsc(x)

    . . . . . . . . . . . . . . cos(x)
    And we have: . .1 - -------- . = . 1 - cot(x)
    . . . . . . . . . . . . . . sin(x)

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