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Math Help - What is the process to prove these Identities

  1. #1
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    Question What is the process to prove these Identities

    I would like some input in which identities to use for each problem. I'm never sure whether to use sum or differences, double angle, 1/2 angle, and etc..there are so many, so I guess and check, which isn't the smart way. I would like to know how to choose the identities.

    #1
    cos(4x) = 1 - 2sin^2(2x)

    #2
    sin(x + pi/2)= cos x


    #3
    tan^2(x) = sec^2(x)= sin^2(x) - cos^2(x)

    #4

    cot(x)/(1 + csc(x)) = (csc(x) - 1)/ cot(x)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by yeunju View Post
    I would like some input in which identities to use for each problem. I'm never sure whether to use sum or differences, double angle, 1/2 angle, and etc..there are so many, so I guess and check, which isn't the smart way. I would like to know how to choose the identities.

    #1
    cos(4x) = 1 - 2sin^2(2x)
    This looks like a half angle fromula with 2x instead of x, so try a half angle identity.

    #2
    sin(x + pi/2)= cos x
    Since you have the sign of a sum try the sum identity for sins

    #3
    tan^2(x) = sec^2(x)= sin^2(x) - cos^2(x)
    replace tan and sec with their definitions in terms of sins and cosines and simplify

    #4

    cot(x)/(1 + csc(x)) = (csc(x) - 1)/ cot(x)
    This is equivalent to:

    cot^2(x)=(csc(x)+1)(csc(x)-1)=csc^2(x)-1

    Now replace everything with its definition in terms of sins and cosins and simplify both sides.

    CB
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