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Math Help - trigonometric

  1. #1
    xavior
    Guest

    trigonometric

    To confirm

    1. 5 + sin^2 A=sinA + 5

    2. cos^4 A-sin^4 A=cos2A

    3. cos^2 A (2tan^2 A + sec^2 A)= 2-cos2A
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Quote Originally Posted by xavior
    To confirm

    1. 5 + sin^2 A=sinA + 5

    2. cos^4 A-sin^4 A=cos2A

    3. cos^2 A (2tan^2 A + sec^2 A)= 2-cos2A
    1. 5 + sin^2 A=sinA + 5 ----------------------------(1)
    When in doubt, put a numerical value to the angle.
    Say angle A = 30deg
    5 +sin^2(30deg) =? sin(30deg) +5
    5 +(0.5)^2 =? 0.5 +5
    5.25 =? 5.5
    No.
    So (1) is not true for A=30deg. Therefore, (1) is not confirmed for all values of A.
    Meaning, (1) is not an identity.

    An identity is true for all the values of the angle in question. ---------*****

    ---------------------------------------------------------
    2. cos^4 A-sin^4 A = cos2A ------------(2)

    I tried A=30deg and it proved true, so, maybe (2) is an identity.
    Let us see.

    Lefthand Side, (LHS) =
    = cos^4(A) -sin^4(A)
    Using a^2 -b^2 = (a+b)(a-b),
    = [cos^2(A) +sin^2(A)]*[cos^2(A) -sin^2(A)]
    = [1]*[cos(2A)]
    = cos(2A)
    = Righthand Side, RHS.

    Therefore, confirmed.

    -----------------------------------------------------
    3. cos^2 A (2tan^2 A + sec^2 A)= 2-cos2A

    I tried again A=30deg and (3) was true, so, maybe (3) is an identity.

    LHS =
    = cos^2(A) *[2tan^2(A) +sec^2(A)]
    = 2cos^2(A)*tan^2(A) +cos^2(A)*sec^2(A)
    = 2cos^2(A)[sin2(A)/cos^2(A)] +cos^2(A)[1/cos^2(A)]
    = 2sin^2(A) +1
    = 2[1 -cos^2(A)] +1
    = 2 -2cos^2(A) +1
    = 2 -2cos^2(A) +[sin^2(A) +cos^2(A)]
    = 2 -cos^2(A) +sin^2(A)
    = 2 -[cos^2(A) -sin^2(A)]
    = 2 -[cos(2A)]
    = 2 -cos(2A)
    = RHS

    Therefore, confirmed.

    --------------------------------------
    We used the trig identities:
    ---sin^2(A) +cos^2(A) = 1
    ---cos(2A) = cos^2(A) -sin^2(A)
    ---tanA = sinA/cosA
    ---secA = 1/cosA
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